On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
The generation of Gaussian random fields over a physical domain is a challenging problem in computational mathematics, especially when the correlation length is short and the field is rough. The traditional approach is to make use of a truncated Karhunen-Loeve (KL) expansion, but the generation of even a single realisation of the field may then be effectively beyond reach (especially for 3-dimensional domains) if the need is to obtain an expected L2 error of say 5%, because of the potentially very slow convergence of the KL expansion. In this talk, based on joint work with Ivan Graham, Frances Kuo, Dirk Nuyens, and Rob Scheichl, a completely different approach is used, in which the field is initially generated at a regular grid on a 2- or 3-dimensional rectangle that contains the physical domain, and then possibly interpolated to obtain the field at other points. In that case there is no need for any truncation. Rather the main problem becomes the factorisation of a large dense matrix. For this we use circulant embedding and FFT ideas. Quasi-Monte Carlo integration is then used to evaluate the expected value of some functional of the finite-element solution of an elliptic PDE with a random field as input.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
The generation of Gaussian random fields over a physical domain is a challenging problem in computational mathematics, especially when the correlation length is short and the field is rough. The traditional approach is to make use of a truncated Karhunen-Loeve (KL) expansion, but the generation of even a single realisation of the field may then be effectively beyond reach (especially for 3-dimensional domains) if the need is to obtain an expected L2 error of say 5%, because of the potentially very slow convergence of the KL expansion. In this talk, based on joint work with Ivan Graham, Frances Kuo, Dirk Nuyens, and Rob Scheichl, a completely different approach is used, in which the field is initially generated at a regular grid on a 2- or 3-dimensional rectangle that contains the physical domain, and then possibly interpolated to obtain the field at other points. In that case there is no need for any truncation. Rather the main problem becomes the factorisation of a large dense matrix. For this we use circulant embedding and FFT ideas. Quasi-Monte Carlo integration is then used to evaluate the expected value of some functional of the finite-element solution of an elliptic PDE with a random field as input.
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
Conformable Chebyshev differential equation of first kindIJECEIAES
In this paper, the Chebyshev-I conformable differential equation is considered. A proper power series is examined; there are two solutions, the even solution and the odd solution. The Rodrigues’ type formula is also allocated for the conformable Chebyshev-I polynomials.
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSTahia ZERIZER
In this article, we study boundary value problems of a large
class of non-linear discrete systems at two-time-scales. Algorithms are given to implement asymptotic solutions for any order of approximation.
A crystallographic group is a group acting on R^n that contains a translation subgroup Z^n as a finite index subgroup. Here we consider which Coxeter groups are crystallographic groups. We also expose the enumeration in dimension 2 and 3. Then we shortly give the principle under which the enumeration of N dimensional crystallographic groups is done.
The Probability that a Matrix of Integers Is DiagonalizableJay Liew
The Probability that a
Matrix of Integers Is Diagonalizable
Andrew J. Hetzel, Jay S. Liew, and Kent E. Morrison
1. INTRODUCTION. It is natural to use integer matrices for examples and exercises
when teaching a linear algebra course, or, for that matter, when writing a textbook in
the subject. After all, integer matrices offer a great deal of algebraic simplicity for particular
problems. This, in turn, lets students focus on the concepts. Of course, to insist
on integer matrices exclusively would certainly give the wrong idea about many important
concepts. For example, integer matrices with integer matrix inverses are quite
rare, although invertible integer matrices (over the rational numbers) are relatively
common. In this article, we focus on the property of diagonalizability for integer matrices
and pose the question of the likelihood that an integer matrix is diagonalizable.
Specifically, we ask: What is the probability that an n × n matrix with integer entries is
diagonalizable over the complex numbers, the real numbers, and the rational numbers,
respectively?
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Density theorems for Euclidean point configurations
1. Density theorems for Euclidean point
configurations
Vjekoslav Kovač (University of Zagreb)
Joint work with P. Durcik, K. Falconer, L. Rimanić, and A. Yavicoli
Supported by HRZZ UIP-2017-05-4129 (MUNHANAP)
Croatian–German meeting on analysis
and mathematical physics
March 25, 2021
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2. Euclidean density theorems
There are patterns in large but otherwise arbitrary structures!
The main idea behind Ramsey theory (⊆ combinatorics), but also
widespread in other areas of mathematics.
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3. Euclidean density theorems
Euclidean density theorems could belong to:
• geometric measure theory (28A12, etc.);
• Ramsey theory (05D10);
• arithmetic combinatorics (11B25, 11B30, etc.).
Harmonic analysis seems to be the most powerful tool for attacking
this type of problems.
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4. Density theorems study “large” measurable sets
When is a measurable set A considered large?
• For A ⊆ [0, 1]d this means
|A| > 0
(the Lebesgue measure).
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5. Density theorems study “large” measurable sets
When is a measurable set A considered large?
• For A ⊆ Rd this means
δ(A) := lim sp
R→∞
sp
x∈Rd
|A ∩ (x + [0, R]d)|
Rd
> 0
(the upper Banach density).
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6. Classical results
A question by Székely (1982)
For every measurable set A ⊆ R2 satisfying δ(A) > 0 is there a
number λ0 = λ0(A) such that for each λ ∈ [λ0, ∞) there exist
points x, x0 ∈ A satisfying |x − x0| = λ?
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7. Classical results
A question by Székely (1982)
For every measurable set A ⊆ R2 satisfying δ(A) > 0 is there a
number λ0 = λ0(A) such that for each λ ∈ [λ0, ∞) there exist
points x, x0 ∈ A satisfying |x − x0| = λ?
Yes:
Furstenberg, Katznelson, and Weiss (1980s);
Falconer and Marstrand (1986);
Bourgain (1986).
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8. Classical results — simplices
Δ = the set of vertices of a non-degenerate n-dimensional simplex
Theorem (Bourgain (1986))
For every measurable set A ⊆ Rn+1 satisfying δ(A) > 0 there is a
number λ0 = λ0(A, Δ) such that for each λ ∈ [λ0, ∞) the set A
contains an isometric copy of λΔ.
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9. Classical results — simplices
Δ = the set of vertices of a non-degenerate n-dimensional simplex
Theorem (Bourgain (1986))
For every measurable set A ⊆ Rn+1 satisfying δ(A) > 0 there is a
number λ0 = λ0(A, Δ) such that for each λ ∈ [λ0, ∞) the set A
contains an isometric copy of λΔ.
Alternative proofs by Lyall and Magyar (2016, 2018, 2019).
Open question #1 (Bourgain?)
When n ≥ 2, does the same hold for subsets of Rn?
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10. Classical results
Bourgain (1986) also gave a counterexample for 3-term arithmetic
progressions, i.e., isometric copies of dilates of {0, 1, 2}.
A :=
{︀
x ∈ Rd
: (∃m ∈ Z)
(︀
m − ϵ < |x|2
< m + ϵ
)︀}︀
10/63
11. Classical results
Bourgain (1986) also gave a counterexample for 3-term arithmetic
progressions, i.e., isometric copies of dilates of {0, 1, 2}.
A :=
{︀
x ∈ Rd
: (∃m ∈ Z)
(︀
m − ϵ < |x|2
< m + ϵ
)︀}︀
• Take some 0 < ϵ < 1/8. We have δ(A) = 2ϵ > 0.
• The parallelogram law:
|x|2
− 2|x + y|2
+ |x + 2y|2
= |y|2
.
• x, x + y, x + 2y ∈ A
dist(|x|2, Z), dist(|x + y|2, Z), dist(|x + 2y|2, Z) < ϵ
=⇒ dist(2|y|2, Z) < 4ϵ < 1/2
=⇒ not all large numbers are attained by |y|
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12. Generalizations to other point configurations
Open question #2 (Graham? Furstenberg?)
Which point configurations P have the following property: for some
(sufficiently large) dimension d and every measurable A ⊆ Rd with
δ(A) > 0 there exists λ0 = λ0(P, A) ∈ (0, ∞) such that for every
λ ≥ λ0 the set A contains an isometric copy of λP?
The most general known positive result is due to Lyall and Magyar
(2019): this holds for products of vertex-sets of nondegenerate
simplices Δ1 × · · · × Δm.
The most general known negative result is due to Graham (1993): this
fails for configurations that cannot be inscribed in a sphere.
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13. Compact formulations
Δ = the set of vertices of a non-degenerate n-dimensional simplex
Theorem (simplices — compact formulation, Bourgain (1986))
Take δ ∈ (0, 1/2], A ⊆ [0, 1]n+1 measurable, |A| ≥ δ.
Then the set of “scales”
{λ ∈ (0, 1] : A contains an isometric copy of λΔ}
contains an interval of length at least
(︀
exp(δ−C(Δ,n))
)︀−1
.
Such a formulation is qualitatively weaker, but it is quantitative.
Such formulations initiate the race to find better dependencies on δ.
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14. General scheme of the approach
Abstracted from: Cook, Magyar, and Pramanik (2017)
N 0
λ
= configuration “counting” form, identifies the configuration
associated with the parameter λ > 0 (i.e., of “size” λ)
N ϵ
λ
= smoothened counting form; the picture is blurred up to scale
0 < ϵ ≤ 1
The largeness–smoothness multiscale approach:
• λ = scale of largeness
• ϵ = scale of smoothness
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15. General scheme of the approach (continued)
Decompose:
N 0
λ
= N 1
λ
+
(︀
N ϵ
λ
− N 1
λ
)︀
+
(︀
N 0
λ
− N ϵ
λ
)︀
.
N 1
λ
= structured part,
N ϵ
λ
− N 1
λ
= error part,
N 0
λ
− N ϵ
λ
= uniform part.
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16. General scheme of the approach (continued)
For the structured part N 1
λ
we need a lower bound
N 1
λ
≥ c(δ)
that is uniform in λ, but this should be a simpler/smoother problem.
For the uniform part N 0
λ
− N ϵ
λ
we want
lim
ϵ→0
⃒
⃒N 0
λ
− N ϵ
λ
⃒
⃒ = 0
uniformly in λ; this usually leads to some oscillatory integrals.
For the error part N ϵ
λ
− N 1
λ
one tries to prove
J
∑︁
j=1
⃒
⃒N ϵ
λj
− N 1
λj
⃒
⃒ ≤ C(ϵ)o(J)
for lacunary scales λ1 < · · · < λJ; this usually leads to some
multilinear singular integrals. 16/63
17. General scheme of the approach (continued)
We argue by contradiction. Take sufficiently many lacunary scales
λ1 < · · · < λJ such that N 0
λj
= 0 for each j.
The structured part
N 1
λj
≥ c(δ)
dominates the uniform part
⃒
⃒N 0
λj
− N ϵ
λj
⃒
⃒ 1 (for sufficiently small ϵ)
and the error part
⃒
⃒N ϵ
λj
− N 1
λj
⃒
⃒ C(ϵ) (for some j by pigeonholing)
for at least one index j. This contradicts N 0
λj
= 0.
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18. 1. Rectangular boxes
Newer results... a warm-up example
= the set of vertices of an n-dimensional rectangular box
Theorem (Lyall and Magyar (2019))
For every measurable set A ⊆ R2 × · · · × R2 = (R2)n satisfying
δ(A) 0 there is a number λ0 = λ0(A, ) such that for each
λ ∈ [λ0, ∞) the set A contains an isometric copy of λ with sides
parallel to the distinguished 2-dimensional coordinate planes.
Previous particular cases by:
Lyall and Magyar (2016), for n = 2;
Durcik and K. (2018), general n, but in (R5)n.
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19. 1. Rectangular boxes — quantitative strengthening
Fix b1, . . . , bn 0 (box sidelengths).
Theorem (Durcik and K. (2020))
For 0 δ ≤ 1/2 and measurable A ⊆ ([0, 1]2)n with |A| ≥ δ there
exists an interval I = I(A, b1, . . . , bn) ⊆ (0, 1] of length at least
(︀
exp(δ−C(n)
)
)︀−1
s. t. for every λ ∈ I one can find x1, . . . , xn, y1, . . . , yn ∈ R2 satisfying
(x1 + r1y1, x2 + r2y2, . . . , xn + rnyn) ∈ A for (r1, . . . , rn) ∈ {0, 1}n
;
|yi| = λbi for i = 1, . . . , n.
This improves the bound of Lyall and Magyar (2019) of the form
(︀
exp(exp(· · · exp(C(n)δ−3·2n
) · · · ))
)︀−1
(a tower of height n). 19/63
20. 1. Rectangular boxes
σ = circle measure in R2, f = 1A
Configuration counting form:
N 0
λ
(f) :=
ˆ
(R2)2n
(︁ ∏︁
(r1,...,rn)∈{0,1}n
f(x1+r1y1, . . . , xn+rnyn)
)︁(︁ n
∏︁
k=1
dxk dσλbk
(yk)
)︁
N ϵ
λ
can be obtained by “heating up” N 0
λ
.
g = standard Gaussian, k = Δg.
The approach benefits from the heat equation
∂
∂t
(︀
gt(x)
)︀
=
1
2πt
kt(x).
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21. 1. Rectangular boxes
Smoothened counting form:
N ϵ
λ
(f) :=
ˆ
(R2)2n
(︁
· · ·
)︁(︁ n
∏︁
k=1
(σ ∗ gϵ)λbk
(yk) dxk dyk
)︁
=
ˆ
(R2)2n
F (x)
(︁ n
∏︁
k=1
(σ ∗ gϵ)λbk
(x0
k
− x1
k
)
)︁
dx
F (x) :=
∏︁
(r1,...,rn)∈{0,1}n
f(xr1
1
, . . . , xrn
n
), dx := dx0
1
dx1
1
dx0
2
dx1
2
· · · dx0
n
dx1
n
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22. 1. Rectangular boxes
It is sufficient to show
N 1
λ
(1A) δ2n
,
J
∑︁
j=1
⃒
⃒N ϵ
λj
(1A) − N 1
λj
(1A)
⃒
⃒ . ϵ−C
,
⃒
⃒N 0
λ
(1A) − N ϵ
λ
(1A)
⃒
⃒ . ϵc
for each scale λ 0 and lacunary scales λ1 · · · λJ.
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23. 1. Rectangular boxes — structured part
Partition “most” of the cube ([0, 1]2)n into rectangular boxes
Q1 × · · · × Qn, where
Qk = [lλbk, (l + 1)λbk) × [l0
λbk, (l0
+ 1)λbk)
We only need the box–Gowers–Cauchy–Schwarz inequality:
Q1×Q1×···×Qn×Qn
F (x) dx ≥
(︁
Q1×...×Qn
f
)︁2n
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24. 1. Rectangular boxes — error part
N α
λ
(f) − N β
λ
(f) =
n
∑︁
m=1
L α,β,m
λ
(f)
L α,β,m
λ
(f) := −
1
2π
ˆ β
α
ˆ
(R2)2n
F (x) (σ ∗ kt)λbm (x0
m
− x1
m
)
×
(︁ ∏︁
1≤k≤n
k6=m
(σ ∗ gt)λbk (x0
k
− x1
k
)
)︁
dx
dt
t
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25. 1. Rectangular boxes — error part (continued)
From
∑︀J
j=1
⃒
⃒N ϵ
λj
(1B) − N 1
λj
(1B)
⃒
⃒ we are lead to study
ΘK((fr1,...,rn )(r1,...,rn)∈{0,1}n )
:=
ˆ
(Rd)2n
∏︁
(r1,...,rn)∈{0,1}n
fr1,...,rn (x1 + r1y1, . . . , xn + rnyn)
)︁
K(y1, . . . , yn)
(︁ n
∏︁
k=1
dxk dyk
)︁
Entangled multilinear singular integral forms with cubical structure:
Durcik (2014); K. (2010); Durcik and Thiele (2018: entangled
Brascamp–Lieb).
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26. 1. Rectangular boxes — uniform part
Split F = F 0(x0
1
)F 0(x1
1
).
The uniform part boils down to the decay of ̂︀
σ:
⃒
⃒̂︀
σ(ξ)
⃒
⃒ . (1 + |ξ|)−1/2
.
ˆ
(R2)2
F (σλ ∗ ktλ)(x0
1
− x1
1
) dx0
1
dx1
1
=
ˆ
R2
(F 0
∗ σλ ∗ ktλ)(x0
1
) F 0
(x0
1
) dx0
1
=
ˆ
R2
(
F 0 ∗ σλ ∗ ktλ)(ξ) ̂︁
F 0(ξ) dξ
=
ˆ
R2
⃒
⃒ ̂︁
F 0(ξ)
⃒
⃒2
̂︀
σ(λξ) ̂︀
k(tλξ) dξ
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27. 2. Anisotropic configurations
Polynomial generalizations?
• There are no triangles with sides λ, λ2, and λ3 for large λ
• One can look for triangles with sides λ, λ2 and a fixed angle θ
between them.
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28. 2. Anisotropic configurations
We will be working with anisotropic power-type scalings
(x1, . . . , xn) 7→ (λa1
b1x1, . . . , λan
bnxn).
Here a1, a2, . . . , an, b1, b2, . . . , bn 0 are fixed parameters.
Crucial observation: the heat equation is unaffected by a power-type
change of the time variable
∂
∂t
(︀
gtab(x)
)︀
=
a
2πt
ktab(x).
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29. 2.1 Anisotropic right-angled simplices
Theorem (K. (2020))
For every measurable set A ⊆ Rn+1 satisfying δ(A) 0 there is a
number λ0 = λ0(A, a1, . . . , an, b1, . . . , bn) such that for each
λ ∈ [λ0, ∞) one can find a point x ∈ Rn+1 and mutually orthogonal
vectors y1, y2, . . . , yn ∈ Rn+1 satisfying
{x, x + y1, x + y2, . . . , x + yn} ⊆ A
and
|yi| = λai
bi; i = 1, 2, . . . , n.
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31. 2.1 Anisotropic simplices (continued)
It is sufficient to show:
N 1
λ
(1B) δn+1
Rn+1
,
J
∑︁
j=1
⃒
⃒N ϵ
λj
(1B) − N 1
λj
(1B)
⃒
⃒ . ϵ−C
J1/2
Rn+1
,
⃒
⃒N 0
λ
(1B) − N ϵ
λ
(1B)
⃒
⃒ . ϵc
Rn+1
,
λ 0, J ∈ N, 0 λ1 · · · λJ satisfy λj+1 ≥ 2λj,
R 0 is sufficiently large, 0 δ ≤ 1,
B ⊆ [0, R]n+1 has measure |B| ≥ δRn+1.
(We take B := (A − x) ∩ [0, R]n+1 for appropriate x, R.)
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32. 2.1 Anisotropic simplices — structured part
σH = the spherical measure inside a subspace H.
N ϵ
λ
(f) =
ˆ
(Rn+1)n+1
f(x)
(︁ n
∏︁
k=1
(f ∗ g(ϵλ)ak bk
)(x + yk)
)︁
dσ{y1,...,yn−1}⊥
λan bn
(yn)
dσ{y1,...,yn−2}⊥
λan−1 bn−1
(yn−1) · · · dσ{y1}⊥
λa2 b2
(y2) dσRn+1
λa1 b1
(y1) dx
σH
∗ g ≥
(︁
min
B(0,2)
g
)︁
1B(0,1) φ := |B(0, 1)|−1
1B(0,1)
Bourgain’s lemma (1988):
[0,R]d
f(x)
(︁ n
∏︁
k=1
(f ∗ φtk )(x)
)︁
dx
(︁
[0,R]d
f(x) dx
)︁n+1
.
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33. 2.1 Anisotropic simplices — error part
N α
λ
(f) − N β
λ
(f) =
n
∑︁
m=1
L α,β,m
λ
(f)
L α,β,m
λ
(f) := −
am
2π
ˆ β
α
ˆ
Rn+1
ˆ
SO(n+1,R)
f(x) (f ∗ k(tλ)am bm
)(x + λam
bmUem)
×
(︁ ∏︁
1≤k≤n
k6=m
(f ∗ g(tλ)ak bk
)(x + λak
bkUek)
)︁
dμ(U) dx
dt
t
L α,β,n
λ
(f) = −
an
2π
ˆ β
α
ˆ
(Rn+1)n+1
f(x)
(︁ n−1
∏︁
k=1
(f ∗ g(tλ)ak bk
)(x + yk)
)︁
× (f ∗ k(tλ)an bn
)(x + yn) dσ{y1,...,yn−1}⊥
λan bn
(yn) · · · dσ{y1}⊥
λa2 b2
(y2) dσRn+1
λa1 b1
(y1) dx
dt
t
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34. 2.1 Anisotropic simplices — error part (continued)
Elimination of measures σH:
convolution with a Gaussian . Schwartz tail
. a superposition of dilated Gaussians
The last inequality is the Gaussian domination trick of Durcik (2014).
The same trick also nicely converts the discrete scales λj into
continuous scales.
Not strictly needed here, but convenient for tracking down
quantitative dependence on ϵ.
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35. 2.1 Anisotropic simplices — error part (continued)
From
∑︀J
j=1
⃒
⃒N ϵ
λj
(1B) − N 1
λj
(1B)
⃒
⃒ we are lead to study
ΛK(f0, . . . , fn) :=
ˆ
(Rd)n+1
K(x1 − x0, . . . , xn − x0)
(︁ n
∏︁
k=0
fk(xk) dxk
)︁
Multilinear C–Z operators: Coifman and Meyer (1970s); Grafakos and
Torres (2002).
Here K is a C–Z kernel, but with respect to the quasinorm associated
with our anisotropic dilation structure.
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37. 2.2 Anisotropic boxes
Theorem (K. (2020))
For every measurable set A ⊆ (R2)n satisfying δ(A) 0 there is a
number λ0 = λ0(A, a1, . . . , an, b1, . . . , bn) such that for each
λ ∈ [λ0, ∞) one can find points x1, . . . , xn, y1, . . . , yn ∈ R2 satisfying
{︀
(x1 + r1y1, x2 + r2y2, . . . , xn + rnyn) : (r1, . . . , rn) ∈ {0, 1}n
}︀
⊆ A
and
|yi| = λai
bi; i = 1, 2, . . . , n.
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38. 2.3 Anisotropic trees
T = (V, E) be a finite tree with vertices V and edges E
Theorem (K. (2020))
For every measurable set A ⊆ R2 satisfying δ(A) 0 there is a
number λ0 = λ0(A, T , a1, . . . , an, b1, . . . , bn) such that for each
λ ∈ [λ0, ∞) one can find a set of points
{xv : v ∈ V} ⊆ A
satisfying
|xu − xv| = λak
bk for each edge k ∈ E joining vertices u, v ∈ V.
This is an anisotropic variant of the result by Lyall and Magyar (2018)
on certain distance graphs.
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39. 2.3 Anisotropic configurations — error part
One would like to handle more general distance graphs (generalizing
simplices, boxes, and trees).
For the error part there is a lot of potential in applying entangled
multilinear singular integrals associated with bipartite graphs or
r-partite r-regular hypergraphs.
The only cases studied so far are:
the so-called “twisted paraproduct operator,”
K. (2010.); Durcik and Roos (2018);
the operators with cubical structure,
Durcik (2014, 2015); Durcik and Thiele (2018).
Dyadic models of these operators are significantly easier:
K. (2011); Stipčić (2019).
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40. 3. Arithmetic progressions
Reformulation of Szemerédi’s theorem
For n ≥ 3 and d ≥ 1 there exists a constant C(n, d) such that for
0 δ ≤ 1/2 and a measurable set A ⊆ [0, 1]d with |A| ≥ δ one has
ˆ
[0,1]d
ˆ
[0,1]d
n−1
∏︁
k=0
1A(x + ky) dy dx
≥
⎧
⎨
⎩
(︀
exp(δ−C(n,d))
)︀−1
when 3 ≤ n ≤ 4,
(︀
exp(exp(δ−C(n,d)))
)︀−1
when n ≥ 5.
Follows from the best known type of bounds in Szemerédi’s theorem:
n = 3, log by Heath-Brown (1987)
n = 4, log by Green and Tao (2017)
n ≥ 5, log log by Gowers (2001)
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41. 3. Arithmetic progressions in other ℓp
-norms
Bourgain’s counterexample applies.
Cook, Magyar, and Pramanik (2015) decided to measure gap lengths
in the ℓp-norm for p 6= 1, 2, ∞.
Theorem (Cook, Magyar, and Pramanik (2015))
If p 6= 1, 2, ∞, d sufficiently large, A ⊆ Rd measurable, δ(A) 0,
then ∃λ0 = λ0(p, d, A) ∈ (0, ∞) such that for every λ ≥ λ0 one
can find x, y ∈ Rd satisfying x, x + y, x + 2y ∈ A and kykℓp = λ.
Open question #3 (Cook, Magyar, and Pramanik (2015))
Is it possible to lower the dimensional threshold all the way to d = 2
or d = 3?
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42. 3. Arithmetic progressions
Open question #4 (Durcik, K., and Rimanić)
Prove or disprove: if n ≥ 4, p 6= 1, 2, . . . , n − 1, ∞, d sufficiently
large, A ⊆ Rd measurable, δ(A) 0, then
∃λ0 = λ0(n, p, d, A) ∈ (0, ∞) such that for every λ ≥ λ0 one can
find x, y ∈ Rd satisfying x, x + y, . . . , x + (n − 1)y ∈ A and kykℓp = λ.
It is necessary to assume p 6= 1, 2, . . . , n − 1, ∞.
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43. 3. Arithmetic progressions — compact formulation
Theorem (Durcik and K. (2020))
Take n ≥ 3, p 6= 1, 2, . . . , n − 1, ∞, d ≥ D(n, p), δ ∈ (0, 1/2],
A ⊆ [0, 1]d measurable, |A| ≥ δ. Then the set of ℓp-norms of the
gaps of n-term APs in the set A contains an interval of length at least
⎧
⎨
⎩
(︀
exp(exp(δ−C(n,p,d)))
)︀−1
when 3 ≤ n ≤ 4,
(︀
exp(exp(exp(δ−C(n,p,d))))
)︀−1
when n ≥ 5.
Modifying Bourgain’s example → sharp regarding the values of p
One can take D(n, p) = 2n+3(n + p) → certainly not sharp
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44. 3. Arithmetic progressions
σ(x) = δ(kxkp
ℓp − 1) = a measure supported on the unit sphere in
the ℓp-norm
N 0
λ
(A) :=
ˆ
Rd
ˆ
Rd
n−1
∏︁
k=0
1A(x + ky) dσλ(y) dx
N 0
λ
(A) 0 =⇒ (∃x, y)
(︀
x, x+y, . . . , x+(n−1)y ∈ A, kykℓp = λ
)︀
N ϵ
λ
(A) :=
ˆ
Rd
ˆ
Rd
n−1
∏︁
k=0
1A(x + ky)(σλ ∗ φϵλ)(y) dy dx
for a smooth φ ≥ 0 with
´
Rd φ = 1
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45. 3. Arithmetic progressions — structured part
Proposition handling the structured part
If λ ∈ (0, 1], δ ∈ (0, 1/2], A ⊆ [0, 1]d, |A| ≥ δ, then
N 1
λ
(A) ≥
⎧
⎨
⎩
(︀
exp(δ−C)
)︀−1
when 3 ≤ n ≤ 4,
(︀
exp(exp(δ−C))
)︀−1
when n ≥ 5.
This is essentially the analytical reformulation of Szemerédi’s
theorem (just at scale λ).
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46. 3. Arithmetic progressions — error part
Proposition handling the error part
If J ∈ N, λj ∈ (2−j, 2−j+1] for j = 1, 2, . . . , J, ϵ ∈ (0, 1/2],
A ⊆ [0, 1]d measurable, then
J
∑︁
j=1
⃒
⃒N ϵ
λj
(A) − N 1
λj
(A)
⃒
⃒ ≤ ϵ−C
J1−2−n+2
.
Note the gain of J−2−n+2
over the trivial estimate.
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47. 3. Arithmetic progressions — uniform part
Proposition handling the uniform part
If d ≥ D(n, p), λ, ϵ ∈ (0, 1], A ⊆ [0, 1]d measurable, then
⃒
⃒N 0
λ
(A) − N ϵ
λ
(A)
⃒
⃒ ≤ Cϵ1/3
.
Consequently, the uniform part can be made arbitrarily small by
choosing a sufficiently small ϵ.
Here is where the assumption p 6= 1, 2, . . . , n − 1, ∞ is needed.
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48. 3. Arithmetic progressions — back to the error part
The most interesting part for us is the error part.
We need to estimate
J
∑︁
j=1
κj
(︀
N ϵ
λj
(A) − N 1
λj
(A)
)︀
for arbitrary scales λj ∈ (2−j, 2−j+1] and arbitrary complex signs κj,
with a bound that is sub-linear in J.
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49. 3. Arithmetic progressions — error part
It can be expanded as
ˆ
Rd
ˆ
Rd
n−1
∏︁
k=0
1A(x + ky)K(y) dy dx,
where
K(y) :=
J
∑︁
j=1
κj
(︀
(σλj ∗ φϵλj )(y) − (σλj ∗ φλj )(y)
)︀
is a translation-invariant Calderón–Zygmund kernel.
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50. 3. Arithmetic progressions — error part
If d = 1 and K(y) is a truncation of 1/y, then this becomes the
(dualized and truncated) multilinear Hilbert transform,
ˆ
R
ˆ
[−R,−r]∪[r,R]
n−1
∏︁
k=0
fk(x + ky)
dy
y
dx.
• When n ≥ 4, no Lp-bounds uniform in r, R are known.
• Tao (2016) showed a bound of the form o(J), where
J ∼ log(R/r) is the “number of scales” involved.
• Reproved and generalized by Zorin-Kranich (2016).
• Durcik, K., and Thiele (2016) showed a bound O(J1−ϵ
).
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51. 4. Some other arithmetic configurations
Allowed symmetries play a major role.
Note a difference between:
the so-called corners: (x, y), (x + s, y), (x, y + s) (harder),
isosceles right triangles: (x, y), (x + s, y), (x, y + t)
with kskℓ2 = ktkℓ2 (easier).
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52. 4.1 Corners
Theorem (Durcik, K., and Rimanić (2016))
If p 6= 1, 2, ∞, d sufficiently large, A ⊆ Rd × Rd measurable,
δ(A) 0, then ∃λ0 = λ0(p, d, A) ∈ (0, ∞) such that for every
λ ≥ λ0 one can find x, y, s ∈ Rd satisfying
(x, y), (x + s, y), (x, y + s) ∈ A and kskℓp = λ.
Generalizes the result of Cook, Magyar, and Pramanik (2015) via the
skew projection (x, y) 7→ y − x.
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54. 4.2 AP-extended boxes
Consider the configuration in Rd1 × · · · × Rdn consisting of:
(x1 + k1s1, x2 + k2s2, . . . , xn + knsn), k1, k2, . . . , kn ∈ {0, 1},
(x1+2s1, x2, . . . , xn), (x1, x2+2s2, . . . , xn), . . . , (x1, x2, . . . , xn+2sn).
Fix b1, . . . , bn 0 and p 6= 1, 2, ∞.
Theorem (Durcik and K. (2018))
There exists a dimensional threshold dmin such that for any d1, d2, . . . ,
dn ≥ dmin and any measurable set A with δ(A) 0 one can find
λ0 0 with the property that for any λ ≥ λ0 the set A contains the
above 3AP-extended box with ksikℓp = λbi, i = 1, 2, . . . , n.
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55. 4.3 Corner-extended boxes
Consider the config. in Rd1 × Rd1 × · · · × Rdn × Rdn consisting of:
(x1 + k1s1, . . . , xn + knsn, y1, y2, . . . , yn), k1, k2, . . . , kn ∈ {0, 1},
(x1, x2, . . . , xn, y1+s1, y2, . . . , yn), . . . , (x1, x2, . . . , xn, y1, y2, . . . , yn+sn).
Fix b1, . . . , bn 0 and p 6= 1, 2, ∞.
Theorem (Durcik and K. (2018))
There exists a dimensional threshold dmin such that for any d1, . . . ,
dn ≥ dmin and any measurable set A with δ(A) 0 one can find
λ0 0 with the property that for any λ ≥ λ0 the set A contains the
above corner-extended box with ksikℓp = λbi, i = 1, 2, . . . , n.
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56. 5. Very dense sets
Theorem (Falconer, K., and Yavicoli (2020))
If d ≥ 2 and A ⊆ Rd is measurable with δ(A) 1 − 1
n−1
, then for
every n-point configuration P there exists λ0 0 s. t. for every
λ ≥ λ0 the set A contains an isometric copy of λP.
The result would be trivial for δ(A) 1 − 1
n
and rotations would not
even be needed there.
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57. 5. Very dense sets — lower bound
What can one say about the lower bound for such density threshold
(depending on n)?
Let us return to arithmetic progressions!
Theorem (Falconer, K., and Yavicoli (2020))
For all n, d ≥ 2 there exists a measurable set A ⊆ Rd of density at
least
1 −
10 log n
n1/5
s.t. there are arbitrarily large values of λ for which A contains no
congruent copy of λ{0, 1, . . . , n − 1}.
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58. 5. Very dense sets — lower bound
It is hard to come up with anything better than Bourgain’s
(counter)example.
Take ϵ := 10n−1/5 log n ∈ (0, 1) and define:
A :=
{︀
x ∈ Rd
: dist(|x|2
, Z) (1 − ϵ)/2
}︀
.
It has density 1 − ϵ.
If x0, x1, . . . , xn−1 ∈ A stand in an arithmetic progression, then the
parallelogram law gives
ak+2 − 2ak+1 + ak = 2λ2
for ak = |xk|2.
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59. 5. Very dense sets — lower bound
The main idea is the following:
• On the one hand, the first n terms of (ak) modulo 1 completely
avoid the interval [(1 − ϵ)/2, (1 + ϵ)/2] ⊆ T of length ϵ.
• On the other hand, for sufficiently large n, the first n terms of
the quadratic sequence (ak) should be “sufficiently uniformly
distributed” over T.
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60. 5. Very dense sets — lower bound
The main idea is the following:
• On the one hand, the first n terms of (ak) modulo 1 completely
avoid the interval [(1 − ϵ)/2, (1 + ϵ)/2] ⊆ T of length ϵ.
• On the other hand, for sufficiently large n, the first n terms of
the quadratic sequence (ak) should be “sufficiently uniformly
distributed” over T.
Take λ such that
λ2
∈ −1+
p
5
2
+ Z
which is “badly approximable” by rationals.
Use some inequality for the discrepancy, or the Erdős–Turán
inequality and work out van der Corput’s trick.
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61. 5. Very dense sets
Open question #5 (Falconer, K., and Yavicoli)
What is the smallest 0 ≤ δmin(d, n) 1 such that every measurable
set A ∈ Rd of upper density δmin(d, n) contains all sufficiently
large scale similar copies of all n-point configurations?
Previous results give
1 −
10 log n
n1/5
≤ δmin(d, n) ≤ 1 −
1
n − 1
.
Is it possible to improve either one of the two asymptotic bounds
1 − O(n−1/5 log n) and 1 − O(n−1) as n → ∞?
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62. Conclusion
• The largeness–smoothness multiscale approach is quite flexible.
• Its applicability largely depends on the current state of the art
on estimates for multilinear singular and oscillatory integrals.
• It gives superior quantitative bounds.
• It can be an overkill in relation to problems without any
arithmetic structure — it was primarily devised to handle
arithmetic progressions and similar patterns.
62/63