Past and future of the cap set problem
We survey the history of the capset problem in the context of related results on progression-free sets, discuss recent progress, and mention further directions to explore.
math.CO↗
arXiv subjects
Vsevolod F. Lev
Publications and source records attributed to Vsevolod F. Lev.
Quotient sets in nonabelian groups
We show that for a finite, nonempty subset $A$ of a group, the quotient set $A^{-1}A:=\{a_1^{-1}a_2\colon a_1,a_2\in A\}$ has size $|A^{-1}A|\ge\frac53\,|A|$, unless $A$ is densely contained in a coset, or in a union of two cosets of a finite subgroup.
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Sum-free sets in $Z_5^n$
It is well-known that for a prime $p\equiv 2\pmod 3$ and integer $n\ge 1$, the maximum possible size of a sum-free subset of the elementary abelian group $\mathbb Z_p^n$ is $\frac13\,(p+1)p^{n-1}$. We establish a matching stability result in the case $p=5$: if $A\subseteq\mathbb Z_5^n$ is a sum-free subset of size $|A|>\frac32\cdot5^{n-1}$, then there are a subgroup $H<\mathbb Z_5^n$ of size $|H|=5^{n-1}$ and an element $e\notin H$ such that $A\subseteq(e+H)\cup(-e+H)$.
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Towards $3n-4$ in groups of prime order
We show that if $A$ is a subset of a group of prime order $p$ such that $|2A|<2.7652|A|$ and $|A|<1.25\cdot10^{-6}p$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms, and $2A$ contains an arithmetic progression with the same difference and at least $2|A|-1$ terms. This improves a number of previously known results.
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A nonlinear bound for the number of subsequence sums
We show that a finite zero-sum-free sequence $α$ over an abelian group has at least $c|α|^{4/3}$ distinct subsequence sums, unless $α$ is "controlled" by a small number of its terms; here $|α|$ denotes the number of terms of $α$, and $c>0$ is an absolute constant.
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The popularity gap
Suppose that $A$ is a finite, nonempty subset of a cyclic group of either infinite or prime order. We show that if the difference set $A-A$ is ``not too large'', then there is a nonzero group element with at least as many as $(2+o(1))|A|^2/|A-A|$ representations as a difference of two elements of $A$; that is, the second largest number of representations is, essentially, twice the average. Here the coefficient $2$ is the best possible. We also prove continuous and multidimensional versions of this result, and obtain similar results for sufficiently dense subsets of an arbitrary abelian group.
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Zero-sum-free sequences with few subsequence sums
We show that a zero-sum-free sequence of length $n$ over an abelian group spans at least $2n$ distinct subsequence sums, unless it possesses a rigid, easily-described structure.
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The structure of higher sumsets
Merging together a result of Nathanson from the early 70s and a recent result of Granville and Walker, we show that for any finite set $A$ of integers with $\min(A)=0$ and $\gcd(A)=1$ there exist two sets, the "head" and the "tail", such that if $m\ge\max(A)-|A|+2$, then the $m$-fold sumset $mA$ consists of the union of these sets and a long block of consecutive integers separating them. We give sharp estimates for the length of the block, and investigate the corresponding stability problem classifying those sets $A$ for which the bound $\max(A)-|A|+2$ cannot be substantially improved.
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Sum-full sets are not zero-sum-free
Let $A$ be a finite, nonempty subset of an abelian group. We show that if every element of $A$ is a sum of two other elements, then $A$ has a nonempty zero-sum subset. That is, a (finite, nonempty) sum-full subset of an abelian group is not zero-sum-free.
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Small doubling in groups with moderate torsion
We determine the structure of a finite subset $A$ of an abelian group given that $|2A|<3(1-ε)|A|$, $ε>0$; namely, we show that $A$ is contained either in a "small" one-dimensional coset progression, or in a union of fewer than $ε^{-1}$ cosets of a finite subgroup. The bounds $3(1-ε)|A|$ and $ε^{-1}$ are best possible in the sense that none of them can be relaxed without tightened another one, and the estimate obtained for the size of the coset progression containing $A$ is sharp. In the case where the underlying group is infinite cyclic, our result reduces to the well-known Freiman's $(3n-3)$-theorem; the former thus can be considered as an extension of the latter onto arbitrary abelian groups, provided that there is "not too much torsion involved".
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Small doubling in cyclic groups
We give a comprehensive description of the sets $A$ in finite cyclic groups such that $|2A|<\frac94|A|$; namely, we show that any set with this property is densely contained in a (one-dimensional) coset progression. This improves earlier results of Deshouillers-Freiman and Balasubramanian-Pandey.
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Small doubling in prime-order groups: from $2.4$ to $2.6$
Improving upon the results of Freiman and Candela-Serra-Spiegel, we show that for a non-empty subset $A\subseteq\mathbb F_p$ with $p$ prime and $|A|<0.0045p$, (i) if $|A+A|<2.59|A|-3$ and $|A|>100$, then $A$ is contained in an arithmetic progression of size $|A+A|-|A|+1$, and (ii) if $|A-A|<2.6|A|-3$, then $A$ is contained in an arithmetic progression of size $|A-A|-|A|+1$. The improvement comes from using the properties of higher energies.
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Point distribution and perfect directions in $F_p^2$
Let $p\ge 3$ be a prime, $S\subseteq\mathbb F_p^2$ a nonempty set, and $w\colon\mathbb F_p^2\to\mathbb R$ a function with $\mathrm{supp}\, w=S$. Applying an uncertainty inequality due to András Biró and the present author, we show that there are at most $\frac12|S|$ directions in $\mathbb F_p^2$ such that for every line $l$ in any of these directions, one has $$ \sum_{z\in l} w(z) = \frac1p\sum_{z\in\mathbb F_p^2} w(z), $$ except if $S$ itself is a line and $w$ is constant on $S$ (in which case all, but one direction have the property in question). The bound $\frac12|S|$ is sharp. As an application, we give a new proof of a result of Rédei-Megyesi about the number of directions determined by a set in a finite affine plane.
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Uncertainty in finite planes
We establish a number of uncertainty inequalities for the additive group of a finite affine plane, showing that for $p$ prime, a nonzero function $f\colon\mathbb F_p^2\to\mathbb C$ and its Fourier transform $\hat f\colon\widehat{\mathbb F_p^2}\to\mathbb C$ cannot have small supports simultaneously. The "baseline" of our investigation is the well-known Meshulam's bound, which we sharpen, for the particular groups under consideration, taking into account not only the sizes of the support sets $\mathrm{supp}\,f$ and $\mathrm{supp}\,\hat f$, but also their structure. Our results imply in particular that, with some explicitly classified exceptions, one has $|\mathrm{supp}\,f||\mathrm{supp}\,\hat f|\ge3p(p-2)$; in comparison, the classical uncertainty inequality gives $|\mathrm{supp}\,f||\mathrm{supp}\,\hat f|\ge p^2$.
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On Isoperimetric Stability
We show that a non-empty subset of an abelian group with a small edge boundary must be large; in particular, if $A$ and $S$ are finite, non-empty subsets of an abelian group such that $S$ is independent, and the edge boundary of $A$ with respect to $S$ does not exceed $(1-γ)|S||A|$ with a real $γ\in(0,1]$, then $|A| \ge 4^{(1-1/d)γ|S|}$, where $d$ is the smallest order of an element of $S$. Here the constant $4$ is best possible. As a corollary, we derive an upper bound for the size of the largest independent subset of the set of popular differences of a finite subset of an abelian group. For groups of exponent $2$ and $3$, our bound translates into a sharp estimate for the additive dimension of the popular difference set. We also prove, as an auxiliary result, the following estimate of possible independent interest: if $A \subset \mathbb Z^n$ is a finite, non-empty downset then, denoting by $w(a)$ the number of non-zero components of the vector $a\in A$, we have \[\frac1{|A|} \sum_{a\in A} w(a) \le \frac12\, \log_2 |A|.\]
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Minimizing the sum of projections of a finite set
Consider the projections of a finite set $A\subset R^n$ onto the coordinate hyperplanes. How small can the sum of the sizes of these projections be, given the size of $A$? In a different form, this problem has been studied earlier in the context of edge-isoperimetric inequalities on graphs, and it is can be derived from the known results that there is a linear order on the set of $n$-tuples with non-negative integer coordinates, such that the sum in question is minimised for the initial segments with respect to this order. We present a new, self-contained and constructive proof, enabling us to obtain a stability result and establish algebraic properties of the smallest possible projection sum. We also solve the problem of minimising the sum of the sizes of the one-dimensional projections.
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Stability result for sets with $3A\ne{\mathbb Z}_5^n$
As an easy corollary of Kneser's Theorem, if $A$ is a subset of the elementary abelian group ${\mathbb Z}_5^n$ of density $5^{-n}|A|>0.4$, then $3A={\mathbb Z}_5^n$. We establish the complementary stability result: if $5^{-n}|A|>0.3$ and $3A\ne{\mathbb Z}_5^n$, then $A$ is contained in a union of two cosets of an index-$5$ subgroup of ${\mathbb Z}_5^n$. Here the density bound $0.3$ is sharp. Our argument combines combinatorial reasoning with a somewhat non-standard application of the character sum technique.
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Symmetric Kneser's Theorem with Trios and $3$-Transform
We give a new equivalent restatement and a new proof in terms of trios to the classical Kneser's theorem. In the finite case, our restatement takes the following, particularly symmetric shape: if $A$, $B$, and $C$ are subsets of a finite abelian group $G$ such that $A+B+C\ne G$, then, denoting by $H$ the period of the sumset $A+B+C$, we have $$ |A|+|B|+|C| \le |G|+|H|. $$ The proof is based on an extension of the familiar Dyson transform onto set systems containing three (or more) sets.
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