Integer sets with identical representation functions
We present a versatile construction allowing one to obtain pairs of integer sets with infinite symmetric difference, infinite intersection, and identical representation functions.
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arXiv subjects
Vsevolod F. Lev
Publications and source records attributed to Vsevolod F. Lev.
Quadratic residues and difference sets
It has been conjectured by Sarkozy that with finitely many exceptions, the set of quadratic residues modulo a prime $p$ cannot be represented as a sumset $\{a+b\colon a\in A, b\in B\}$ with non-singleton sets $A,B\subset F_p$. The case $A=B$ of this conjecture has been recently established by Shkredov. The analogous problem for differences remains open: is it true that for all sufficiently large primes $p$, the set of quadratic residues modulo $p$ is not of the form $\{a'-a"\colon a',a"\in A,\,a'\ne a"\}$ with $A\subset F_p$? We attack here a presumably more tractable variant of this problem, which is to show that there is no $A\subset F_p$ such that every quadratic residue has a \emph{unique}representation as $a'-a"$ with $a',a"\in A$, and no non-residue is represented in this form. We give a number of necessary conditions for the existence of such $A$, involving for the most part the behavior of primes dividing $p-1$. These conditions enable us to rule out all primes $p$ in the range $13<p<10^{18}$ (the primes $p=5$ and $p=13$ being conjecturally the only exceptions).
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Discrete norms of a matrix and the converse to the Expander Mixing Lemma
We define the discrete norm of a complex $m\times n$ matrix $A$ by $$ \|A\|_Δ:= \max_{0\neξ\in\{0,1\}^n} \frac{\|Aξ\|}{\|ξ\|}, $$ and show that $$ \frac c{\sqrt{\log h(A)+1}}\,\|A\| \le \|A\|_Δ\le \|A\|, $$ where $c>0$ is an explicitly indicated absolute constant, $h(A)=\sqrt{\|A\|_1\|A\|_\infty}/\|A\|$, and $\|A\|_1,\|A\|_\infty$, and $\|A\|=\|A\|_2$ are the induced operator norms of $A$. Similarly, for the \emph{discrete Rayleigh norm} $$ \|A\|_P := \max_{\substack{0\neξ\in\{0,1\}^m \\ 0\neη\in\{0,1\}^n}} \frac{|ξ^tAη|}{\|ξ\|\|η\|} $$ we prove the estimate $$ \frac c{\log h(A)+1}\,\|A\| \le \|A\|_P \le \|A\|. $$ These estimates are shown to be essentially best possible. As a consequence, we obtain another proof of the (slightly sharpened and generalized version of the) converse to the expander mixing lemma by Bollobas-Nikiforov and Bilu-Linial.
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Flat-containing and shift-blocking sets in $F_2^r$
For non-negative integers $r\ge d$, how small can a subset $C\subset F_2^r$ be, given that for any $v\in F_2^r$ there is a $d$-flat passing through $v$ and contained in $C\cup\{v\}$? Equivalently, how large can a subset $B\subset F_2^r$ be, given that for any $v\in F_2^r$ there is a linear $d$-subspace not blocked non-trivially by the translate $B+v$? A number of lower and upper bounds are obtained.
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Solving $a\pm b=2c$ in the elements of finite sets
We show that if $A$ and $B$ are finite sets of real numbers, then the number of triples $(a,b,c)\in A\times B\times (A\cup B)$ with $a+b=2c$ is at most $(0.15+o(1))(|A|+|B|)^2$ as $|A|+|B|\to\infty$. As a corollary, if $A$ is antisymmetric (that is, $A\cap(-A)=\est$), then there are at most $(0.3+o(1))|A|^2$ triples $(a,b,c)$ with $a,b,c\in A$ and $a-b=2c$. In the general case where $A$ is not necessarily antisymmetric, we show that the number of triples $(a,b,c)$ with $a,b,c\in A$ and $a-b=2c$ is at most $(0.5+o(1))|A|^2$. These estimates are sharp.
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Edge-isoperimetric problem for Cayley graphs and generalized Takagi function
Let $G$ be a finite abelian abelian group of exponent $m\ge 2$. For subsets $A,S\subset G$, denote by $\partial_S(A)$ the number of edges from $A$ to its complement $G\setminus A$ in the directed Cayley graph, induced by $S$ on $G$. We show that if $S$ generates $G$, and $A$ is non-empty, then $$ \partial_S(A) \ge \frac{e}m\,|A|\ln\frac{|G|}{|A|}. $$ Here the coefficient $e=2.718...$ is best possible and cannot be replaced with a number larger than $e$. For homocyclic groups $G$ of exponent $m$, we find an explicit closed-form expression for $\partial_S(A)$ in the case where $S$ is a "standard" generating subset of $G$, and $A$ is an initial segment of $G$ with respect to the lexicographic order, induced by $S$ on $G$. Namely, we show that in this situation $$ \partial_S(A) = |G|\,ω_m(|A|/|G|), $$ where $ω_2$ is the Takagi function, and $ω_m$ for $m\ge 3$ is an appropriate generalization thereof. This particular case is of special interest, since for $m\in\{2,3,4\}$ it is known to yield the smallest possible value of $\partial_S(A)$, over all sets $A\subset G$ of given size. We give this classical result a new proof, somewhat different from the standard one. We also give a new, short proof of the Boros-Pales inequality $$ω_2(\frac{x+y}2) \le \frac{ω_2(x) + ω_2(y)}2 + \frac12\,|y-x|, $$ establish an extremal characterization of the Takagi function as the (pointwise) maximal function, satisfying this inequality and the boundary condition $\max\{ω_2(0),ω_2(1)\}\le 0$, and obtain similar results for the 3-adic analog $ω_3$ of the Takagi function.
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Small asymmetric sumsets in elementary abelian 2-groups
Let A and B be subsets of an elementary abelian 2-group G, none of which are contained in a coset of a proper subgroup. Extending onto potentially distinct summands a result of Hennecart and Plagne, we show that if |A+B|<|A|+|B|, then either A+B=G, or the complement of A+B in G is contained in a coset of a subgroup of index at least 8, whence |A+B| is at least 7/8 |G|. We indicate conditions for the containment to be strict, and establish a refinement in the case where the sizes of A and B differ significantly.
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The Largest Eigenvalue and Bi-Average Degree of a Graph
We show that for a graph $G$ with the vertex set $V$ and the largest eigenvalue $λ_{\max}(G)$, letting $$ M(G) := \max_{X,Y \subset V} \frac{e(X,Y)}{\sqrt{|X||Y|}} $$ (where $e(X,Y)$ denotes the number of edges between $X$ and $Y$), we have $$ M(G) \le λ_{\max}(G) \le \big(\frac14 \log|V| + 1 \big) \M(G). $$ Here the lower bound is attained if $G$ is regular or bi-regular, whereas the logarithmic factor in the upper bound, conjecturally, can be improved --- although we present an example showing that it cannot be replaced with a factor growing slower than $(\log |V|/\log\log|V|)^{1/8}$. Further refinements are established, particularly in the case where $G$ is bipartite.
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On the size of dissociated bases
We prove that the sizes of the maximal dissociated subsets of a given finite subset of an abelian group differ by a logarithmic factor at most. On the other hand, we show that the set $\{0,1\}^n\seq\Z^n$ possesses a dissociated subset of size $\Ome(n\log n)$; since the standard basis of $\Z^n$ is a maximal dissociated subset of $\{0,1\}^n$ of size $n$, the result just mentioned is essentially sharp.
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Kakeya-type sets in finite vector spaces
For a finite vector space $V$ and a non-negative integer $r\le\dim V$ we estimate the smallest possible size of a subset of $V$, containing a translate of every $r$-dimensional subspace. In particular, we show that if $K\subset V$ is the smallest subset with this property, $n$ denotes the dimension of $V$, and $q$ is the size of the underlying field, then for $r$ bounded and $r<n\le rq^{r-1}$ we have $|V\setminus K|=Θ(nq^{n-r+1})$. This improves previously known bounds $|V\setminus K|=Ω(q^{n-r+1})$ and $|V\setminus K|=O(n^2q^{n-r+1})$.
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Translation invariance in groups of prime order
We prove that there is an absolute constant $c>0$ with the following property: if $Z/pZ$ denotes the group of prime order $p$, and a subset $A\subset Z/pZ$ satisfies $1<|A|<p/2$, then for any positive integer $m<\min\{c|A|/\ln|A|,\sqrt{p/8}\}$ there are at most $2m$ non-zero elements $b\in Z/pZ$ with $|(A+b)\setminus A|\le m$. This (partially) extends onto prime-order groups the result, established earlier by S. Konyagin and the present author for the group of integers. We notice that if $A\subset Z/pZ$ is an arithmetic progression and $m<|A|<p/2$, then there are exactly $2m$ non-zero elements $b\in Z/pZ$ with $|(A+b)\setminus A|\le m$. Furthermore, the bound $c|A|/\ln|A|$ is best possible up to the value of the constant $c$. On the other hand, it is likely that the assumption $m<\sqrt{p/8}$ can be dropped or substantially relaxed.
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The continuous postage stamp problem
For a real set $A$ consider the semigroup $S(A)$, additively generated by $A$; that is, the set of all real numbers representable as a (finite) sum of elements of $A$. If $A \subset (0,1)$ is open and non-empty, then $S(A)$ is easily seen to contain all sufficiently large real numbers, and we let $G(A) := \sup \{u \in R \colon u \notin S(A) \}$. Thus, $G(A)$ is the smallest number with the property that any $u>G(A)$ is representable as indicated above. We show that if the measure of $A$ is large, then $G(A)$ is small; more precisely, writing for brevity $α:= \mes A$ we have $$ G(A) \le (1-α) \lfloor 1/α\rfloor \quad &\text{if $0 < α\le 0.1$}, (1-α+α\{1/α\})\lfloor 1/α\rfloor \quad &\text{if $0.1 \le α\le 0.5$}, 2(1-α) \quad &\text{if $0.5 \le α\le 1$}. $$ Indeed, the first and the last of these three estimates are the best possible, attained for $A=(1-α,1)$ and $A=(1-α,1)\setminus\{2(1-α)\}$, respectively; the second is close to the best possible and can be improved by $α\{1/α\} \lfloor 1/α\rfloor \le \{1/α\}$ at most. The problem studied is a continuous analogue of the linear Diophantine problem of Frobenius (in its extremal settings due to Erdos and Graham), also known as the "postage stamp problem" or the "coin exchange problem".
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Generating abelian groups by addition only
We define the positive diameter of a finite group $G$ with respect to a generating set $A\subset G$ to be the smallest non-negative integer $n$ such that every element of $G$ can be written as a product of at most $n$ elements of $A$. This invariant, which we denote by $\diam_A^+(G)$, can be interpreted as the diameter of the Cayley digraph induced by $A$ on $G$. In this paper we study the positive diameters of a finite abelian group $G$ with respect to its various generating sets $A$. More specifically, we determine the maximum possible value of $\diam_A^+(G)$ and classify all generating sets for which this maximum value is attained. Also, we determine the maximum possible cardinality of $A$ subject to the condition that $\diam_A^+(G)$ is "not too small". Conceptually, the problems studied are closely related to our earlier work and the results obtained shed a new light on the subject. Our original motivation came from connections with caps, sum-free sets, and quasi-perfect codes.
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How long does it take to generate a group?
The diameter of a finite group $G$ with respect to a generating set $A$ is the smallest non-negative integer $n$ such that every element of $G$ can be written as a product of at most $n$ elements of $A \cup A^{-1}$. We denote this invariant by $\diam_A(G)$. It can be interpreted as the diameter of the Cayley graph induced by $A$ on $G$ and arises, for instance, in the context of efficient communication networks. In this paper we study the diameters of a finite abelian group $G$ with respect to its various generating sets $A$. We determine the maximum possible value of $\diam_A(G)$ and classify all generating sets for which this maximum value is attained. Also, we determine the maximum possible cardinality of $A$ subject to the condition that $\diam_A(G)$ is "not too small". Connections with caps, sum-free sets, and quasi-perfect codes are discussed.
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Character-free approach to progression-free sets
We present an elementary combinatorial argument showing that the density of a progression-free set in a finite r-dimensional vector space is O(1/r).
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The Erdos-Turan problem in infinite groups
Let $G$ be an infinite abelian group with $|2G|=|G|$. We show that if $G$ is not the direct sum of a group of exponent 3 and the group of order 2, then $G$ possesses a perfect additive basis; that is, there is a subset $S\subseteq G$ such that every element of $G$ is uniquely representable as a sum of two elements of $S$. Moreover, if $G$ \emph{is} the direct sum of a group of exponent 3 and the group of order 2, then it does not have a perfect additive basis; however, in this case there is a subset $S\subseteq G$ such that every element of $G$ has at most two representations (distinct under permuting the summands) as a sum of two elements of $S$. This solves completely the Erdos-Turan problem for infinite groups. It is also shown that if $G$ is an abelian group of exponent 2, then there is a subset $S\subseteq G$ such that every element of $G$ has a representation as a sum of two elements of $S$, and the number of representations of non-zero elements is bounded by an absolute constant.
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Additive Bases in Abelian Groups
Let $G$ be a finite, non-trivial abelian group of exponent $m$, and suppose that $B_1, ..., B_k$ are generating subsets of $G$. We prove that if $k>2m \ln \log_2 |G|$, then the multiset union $B_1\cup...\cup B_k$ forms an additive basis of $G$; that is, for every $g\in G$ there exist $A_1\subset B_1, ..., A_k\subset B_k$ such that $g=\sum_{i=1}^k\sum_{a\in A_i} a$. This generalizes a result of Alon, Linial, and Meshulam on the additive bases conjecture. As another step towards proving the conjecture, in the case where $B_1, ..., B_k$ are finite subsets of a vector space we obtain lower-bound estimates for the number of distinct values, attained by the sums of the form $\sum_{i=1}^k \sum_{a\in A_i} a$, where $A_i$ vary over all subsets of $B_i$ for each $i=1, >..., k$. Finally, we establish a surprising relation between the additive bases conjecture and the problem of covering the vertices of a unit cube by translates of a lattice, and present a reformulation of (the strong form of) the conjecture in terms of coverings.
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Consecutive integers in high-multiplicity sumsets
Sharpening (a particular case of) a result of Szemeredi and Vu and extending earlier results of Sarkozy and ourselves, we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset to contain a block of consecutive integers of length, comparable with the lengths of the set summands. A corollary of our main result is as follows. Let $k,l\ge 1$ and $n\ge 3$ be integers, and suppose that $A_1,...,A_k\subset[0,l]$ are integer sets of size at least $n$, none of which is contained in an arithmetic progression with difference greater than 1. If $k\ge 2\lceil(l-1)/(n-2)\rceil$, then the sumset $A_1+...+A_k$ contains a block of consecutive integers of length $k(n-1)$.
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