# Convolutions of sets with bounded VC-dimension are uniformly continuous

@article{Sisask2018ConvolutionsOS, title={Convolutions of sets with bounded VC-dimension are uniformly continuous}, author={Olof Sisask}, journal={arXiv: Combinatorics}, year={2018} }

We introduce a notion of VC-dimension for subsets of groups, defining this for a set $A$ to be the VC-dimension of the family $\{ A \cap(xA) : x \in A\cdot A^{-1} \}$. We show that if a finite subset $A$ of an abelian group has bounded VC-dimension, then the convolution $1_A*1_{-A}$ is Bohr uniformly continuous, in a quantitatively strong sense. This generalises and strengthens a version of the stable arithmetic regularity lemma of Terry and Wolf in various ways. In particular, it directly…

## 11 Citations

### Structure and regularity for subsets of groups with finite VC-dimension

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Suppose $G$ is a finite group and $A\subseteq G$ is such that $\{gA:g\in G\}$ has VC-dimension strictly less than $k$. We find algebraically well-structured sets in $G$ which, up to a chosen…

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We show that if $G$ is a discrete Abelian group and $A \subset G$ has $\|1_A\|_{B(G)} \leq M$ then $A$ is $O(\exp(\pi M))$-stable in the sense of Terry and Wolf.

### Quantitative structure of stable sets in finite abelian groups

- MathematicsTransactions of the American Mathematical Society
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We prove an arithmetic regularity lemma for stable subsets of finite abelian groups, generalising our previous result for high-dimensional vector spaces over finite fields of prime order. A…

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### Approximate subgroups with bounded VC-dimension

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We combine the fundamental results of Breuillard, Green, and Tao on the structure of approximate groups, together with "tame" arithmetic regularity methods based on work of the authors and Terry, to…

### Almost periodicity and its applications to Roth’s theorem

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We give a self-contained exposition of several aspects of Croot-Sisask almost periodicity, with a special focus on its application to Roth’s theorem. Using almost periodicity, we obtain a bound on…

### On finite sets of small tripling or small alternation in arbitrary groups

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A qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularityLemma for sets of bounded VC-dimension in finite groups of bounded exponent are obtained.

## References

SHOWING 1-10 OF 23 REFERENCES

### A Szemerédi-type regularity lemma in abelian groups, with applications

- Mathematics
- 2003

Abstract.Szemerédi’s regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemerédi’s regularity lemma in the…

### A Probabilistic Technique for Finding Almost-Periods of Convolutions

- Mathematics
- 2010

We introduce a new probabilistic technique for finding ‘almost-periods’ of convolutions of subsets of groups. This gives results similar to the Bogolyubovtype estimates established by Fourier…

### ROTH’S THEOREM FOR FOUR VARIABLES AND ADDITIVE STRUCTURES IN SUMS OF SPARSE SETS

- MathematicsForum of Mathematics, Sigma
- 2016

We show that if $A\subset \{1,\ldots ,N\}$ does not contain any nontrivial solutions to the equation $x+y+z=3w$ , then $$\begin{eqnarray}|A|\leqslant \frac{N}{\exp (c(\log N)^{1/7})},\end{eqnarray}$$…

### VC-sets and generic compact domination

- Mathematics
- 2015

Let X be a closed subset of a locally compact second countable group G whose family of translates has finite VC-dimension. We show that the topological border of X has Haar measure 0. Under an extra…

### Lower bounds of tower type for Szemerédi's uniformity lemma

- Mathematics, Computer Science
- 1997

This paper shows that the bound is necessarily of tower type, obtaining a lower bound given by a tower of 2s of height proportional to $ \log{(1/ \epsilon)} $).

### Efficient arithmetic regularity and removal lemmas for induced bipartite patterns

- Mathematicsdiscrete Analysis
- 2019

Efficient arithmetic regularity and removal lemmas for induced bipartite patterns, Discrete Analysis 2019:3, 14 pp.
This paper provides a common extension of two recent lines of work: the study of…

### Freiman's theorem in an arbitrary abelian group

- Mathematics
- 2005

A famous result of Freiman describes the structure of finite sets A ⊆ ℤ with small doubling property. If |A + A| ⩽ K|A|, then A is contained within a multidimensional arithmetic progression of…

### Green's sumset problem at density one half

- Mathematics
- 2011

We investigate the size of subspaces in sumsets and show two main results. First, if A is a subset of F_2^n with density at least 1/2 - o(n^{-1/2}) then A+A contains a subspace of co-dimension 1.…

### Finite field models in additive combinatories

- MathematicsBCC
- 2005

The study of many problems in additive combinatorics, such as Szemer\'edi's theorem on arithmetic progressions, is made easier by first studying models for the problem in F_p^n for some fixed small…

### New proofs of Plünnecke-type estimates for product sets in groups

- MathematicsComb.
- 2012

A new method to bound the cardinality of product sets in groups and give three applications, including a new proof of a theorem of Tao on triple products, which generalises these inequalities when no assumption on commutativity is made.