Large sets avoiding patterns

Fraser, Robert; Pramanik, Malabika

doi:10.2140/apde.2018.11.1083

Cited by 18 publications

(33 citation statements)

References 17 publications

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“…, x n´1 q. Under the assumption that f is smooth, and satisfies a regularity condition geometrically equivalent to the graph of f being transverse to any axis-oriented hyperplane, we are able to improve the Fourier dimension bound obtained, though not quite enough to match the Hausdorff dimension bound obtained in [4], except in the fairly trivial case where n " 2.…”

Section: Introductionmentioning

confidence: 80%

“…Then E avoids isoceles triangles if and only if γpEq does not contain any three points forming the vertices of an isosceles triangle. In [4], methods are provided to construct sets E Ă r0, 1s with dim H pEq " log 2{ log 3 « 0.63 such that γpEq does not contain any isosceles triangles, but E is not guaranteed to be Salem. We can use Theorem 1.2 to construct Salem sets E Ă r0, 1s with dim F pEq " 4{9 « 0.44.…”

Section: Isosceles Triangles On Curvesmentioning

confidence: 99%

“…Many other approaches to constructing pattern avoiding sets construct explicit pattern avoiding Salem sets by using various queuing techniques. These methods were pioneered in [5], who constructed an explicit set avoiding nontrivial solutions to the equation x 2 ´x1 " x 4 ´x3 , but [4] showed these techniques held for a much more general family of constructions. Other applications of the method are found in [2] and [1].…”

Section: Introductionmentioning

confidence: 99%

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Large Salem Sets Avoiding Nonlinear Configurations

Denson¹

2021

Preprint

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We construct large Salem sets avoiding patterns, complementing previous constructions of pattern avoiding sets with large Hausdorff dimension. For a (possibly uncountable) family of uniformly Lipschitz functions tf i : pT d q n´2 Ñ T d u, we obtain a Salem subset of T d with dimension d{pn ´1q avoiding nontrivial solutions to the equation x n ´xn´1 " f i px 1 , . . . , x n´2 q. For a countable family of smooth functions tf i : pT d q n´1 Ñ T d u satisfying a modest geometric condition, we obtain a Salem subset of T d with dimension d{pn ´3{4q avoiding nontrivial solutions to the equation x n " f px 1 , . . . , x n´1 q. For a set Z Ă T dn which is the countable union of a family of sets, each with lower Minkowski dimension s, we obtain a Salem subset of T d of dimension pdn ´sq{pn ´1{2q whose Cartesian product does not intersect Z except at points with non-distinct coordinates.

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Section: Introductionmentioning

confidence: 80%

Section: Isosceles Triangles On Curvesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Large Salem Sets Avoiding Nonlinear Configurations

Denson¹

2021

Preprint

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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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“…We remark that Máthé [12], and Fraser and Pramanik [6] studied similar problems for non-linear patterns, under certain conditions, but the sets they construct are not of full dimension, and in some particular cases the dimension obtained is optimal (that is to say, in some cases, there is no set of full dimension without the given non-linear patterns). Since we want to study large sets for an arbitrary dimension function h with h ≺ x d , we focus on the case of linear patterns.…”

Section: Remarkmentioning

confidence: 99%