Large sets avoiding linear patterns

Yavicoli, Alexia

doi:10.1090/proc/13959

Cited by 11 publications

(8 citation statements)

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“…We proved [17], improving a result from [9] and the linear case of [12,Theorem 6.1], that there are sets with full Hausdorff dimension avoiding countably many linear patterns (defined as zeros of linear functions) given in advance. Moreover, we got a more general result for dimension functions instead of Hausdorff dimension.…”

Section: Introductionmentioning

confidence: 66%

Patterns in thick compact sets

Yavicoli¹

2019

Preprint

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We introduce a connection between Newhouse thickness and patterns through a variant of Schmidt's game introduced by Broderick, Fishman and Simmons. This yields an explicit, robust and checkable condition that ensures the presence of patterns in compact sets, in particular in Cantor sets. Contents 1. Introduction. 1.1. Examples and Remarks 2. The potential game 3. Guaranteeing affine copies 3.1. Large thickness guarantees affine copies of finite sets 3.2. Large thickness guarantees bilipschitz patterns 4. Guaranteeing patterns of an explicit size 4.1. Large thickness guarantees affine copies of finite sets of an explicit size 4.2. Large thickness guarantees bilipschitz patterns of an explicit size 5. Proof of Theorem 27 References

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Section: Introductionmentioning

confidence: 66%

Patterns in thick compact sets

Yavicoli¹

2019

Preprint

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“…A consequence of the Lebesgue density theorem is that any set E ⊂ R d of positive Lebesgue measure contains a homothetic copy of every finite set at all sufficiently small scales, so it is natural to seek conditions on sets of zero Lebesgue measure form which this remains true. Perhaps the most natural notion of size to consider is Hausdorff dimension but there are constructions (see for example [6,13,14,16,18,21]) which indicate that Hausdorff dimension cannot, in itself, detect the presence or absence of patterns in sets of Lebesgue measure zero, even in the most basic case of points in arithmetic progressions.…”

Section: Introductionmentioning

confidence: 99%

Intersections of thick compact sets in $${\mathbb {R}}^d$$

Falconer

Yavicoli

2022

Math. Z.

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We introduce a definition of thickness in $${\mathbb {R}}^d$$ R d and obtain a lower bound for the Hausdorff dimension of the intersection of finitely or countably many thick compact sets using a variant of Schmidt’s game. As an application we prove that given any compact set in $${\mathbb {R}}^d$$ R d with thickness $$\tau $$ τ , there is a number $$N(\tau )$$ N ( τ ) such that the set contains a translate of all sufficiently small similar copies of every set in $${\mathbb {R}}^d$$ R d with at most $$N(\tau )$$ N ( τ ) elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness.

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“…Keleti [19] has constructed subsets of full dimension on R which contain no 3-term arithmetic progressions (or more generally similar copies of an a priori fixed set with 3 elements). For extensions and generalisations, and more results in this direction, see the papers [24] of Maga, [25] of Máthé, [9] of Denson, Łaba, and Zahl, and [39] of Yavicoli. Quite surprisingly, one may even construct subsets of R with full Hausdorff dimension and Fourier dimension which avoid 3-term arithmetic progressions: this counterintuitive construction is due to Shmerkin [36].…”

Section: Introductionmentioning

confidence: 99%