On Side Lengths of Corners in Positive Density Subsets of the Euclidean Space

Durcik, Polona; Kovač, Vjekoslav; Rimanic, Luka

doi:10.1093/imrn/rnx093

Cited by 18 publications

(33 citation statements)

References 27 publications

(84 reference statements)

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“…Let us also mention that there are many related papers that study lower-dimensional subsets of the Euclidean space [1,6,33,36,38,31,32,59,37], subsets of the multidimensional integer lattice [49,5,46], or patterns with arithmetic structure [44,39,34,51,25,10,17,16,11,24,15]. We do not discuss these types of results here.…”

Section: Overview Of Previous Resultsmentioning

confidence: 99%

Density theorems for anisotropic point configurations

Kovač

2021

Can. J. Math.-J. Can. Math.

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Several results in the existing literature establish Euclidean density theorems of the following strong type. These results claim that every set of positive upper Banach density in the Euclidean space of an appropriate dimension contains isometric copies of all sufficiently large elements of a prescribed family of finite point configurations. So far, all results of this type discussed linear isotropic dilates of a fixed point configuration. In this paper, we initiate the study of analogous density theorems for families of point configurations generated by anisotropic dilations, i.e., families with power-type dependence on a single parameter interpreted as their size. More specifically, we prove nonisotropic power-type generalizations of a result by Bourgain on vertices of a simplex, a result by Lyall and Magyar on vertices of a rectangular box, and a result on distance trees, which is a particular case of the treatise of distance graphs by Lyall and Magyar. Another source of motivation for this paper is providing additional evidence for the versatility of the approach stemming from the work of Cook, Magyar, and Pramanik and its modification used recently by Durcik and the present author. Finally, yet another purpose of this paper is to single out anisotropic multilinear singular integral operators associated with the above combinatorial problems, as they are interesting on their own.

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Section: Overview Of Previous Resultsmentioning

confidence: 99%

Density theorems for anisotropic point configurations

Kovač

2021

Can. J. Math.-J. Can. Math.

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“…, Ω c,C (τ 2 )), (4.19) where L ′ is as in (4.10).This inequality is the reverse inequality of part (8), and in fact it will not be required in the proof of any of our main theorems. However, it will be required in order to analyse A L (τ 1 , τ 2 ) when L has algebraic coefficients (in Appendix E), so we choose to state and prove it here, close to our argument for part (8).…”

Section: Dimension Reductionmentioning

confidence: 97%

Gowers norms control Diophantine inequalities

Walker¹

2017

Trans. Amer. Math. Soc.

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Abstract. A central tool in the study of systems of linear equations with integer coefficients is the Generalised von Neumann Theorem of Green and Tao. This theorem reduces the task of counting the weighted solutions of these equations to that of counting the weighted solutions for a particular family of forms, the Gowers norms f U s+1 [N ] of the weight f . In this paper we consider systems of linear inequalities with real coefficients, and show that the number of solutions to such weighted diophantine inequalities may also be bounded by Gowers norms. Further, we provide a necessary and sufficient condition for a system of real linear forms to be governed by Gowers norms in this way. In a forthcoming paper we will discuss the case in which the weights are unbounded but suitably pseudorandom, with applications to counting the number of solutions to diophantine inequalities over the primes.

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“…At the moment we are not able to prove a result analogous to Theorem 2 for this grid. Larger grids bring further complications: one should first handle longer arithmetic progressions and it is known that additional restrictions on the values of p are needed; see the remarks in [4].…”

Section: D N D Min and Any Measurable Setmentioning

confidence: 99%

Boxes, extended boxes and sets of positive upper density in the Euclidean space

Durcik¹,

Kovač²

2021

Math. Proc. Camb. Phil. Soc.

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We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2 n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

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On Side Lengths of Corners in Positive Density Subsets of the Euclidean Space

Abstract: Abstract. We generalize a result by Cook, Magyar, and Pramanik [3] on three-term arithmetic progressions in subsets of R d to corners in subsets of

Cited by 18 publications

References 27 publications

Density theorems for anisotropic point configurations

Density theorems for anisotropic point configurations

Gowers norms control Diophantine inequalities

Boxes, extended boxes and sets of positive upper density in the Euclidean space

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