The Minkowski sum of two subsets $A$ and $B$ of a finite abelian group $G$ is defined as all pairwise sums of elements of $A$ and $B$: $A + B = \{ a + b : a \in A, b \in B \}$. The largest size of a $(k, \ell)$-sum-free set in $G$ has been of interest for many years and in the case $G = \mathbb{Z}/n\mathbb{Z}$ has recently been computed by Bajnok and Matzke. Motivated by sum-free sets of the torus, Kravitz introduces the noisy Minkowski sum of two sets, which can be thought of as discrete evaluations of these continuous sumsets. That is, given a noise set $C$, the noisy Minkowski sum is defined as $A +_C B = A + B + C$. We give bounds on the maximum size of a $(k, \ell)$-sum-free subset of $\mathbb{Z}/n\mathbb{Z}$ under this new sum, for $C$ equal to an arithmetic progression with common difference relatively prime to $n$ and for any two element set $C$.
We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for this problem we use a stability argument to find the unique extremal graph. We show that our result holds within the larger family of $n$-vertex $k$-chromatic graphs with minimum degree at least $\ell$, again for $n$ sufficiently large. We also maximize the number of independent sets of each fixed size in $n$-vertex 3-chromatic 2-connected graphs. We finally ad…
A minor-model of a graph $H$ in a graph $G$ is a subgraph of $G$ that can be contracted to $H$. We prove that for a positive integer $\ell$ and a non-empty planar graph $H$ with at least $\ell-1$ connected components, there exists a function $f_{H, \ell}:\mathbb{N}\rightarrow \mathbb{R}$ satisfying the property that every graph $G$ with a family of vertex subsets $Z_1, \ldots, Z_m$ contains either $k$ pairwise vertex-disjoint minor-models of $H$ each intersecting at least $\ell$ sets among prescribed vertex sets, or a vertex subset of size at most $f_{H, \ell}(k)$ that meets all such minor-mod…
Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, we construct the first known family of partial difference sets with negative Latin square type parameters in nonelementary abelian groups, the groups $\Z_4^{2k}\times \Z_2^{4 \ell-4k}$ for all $k$ when $\ell$ is odd and for all $k < \ell$ when $\ell$ is even. Similarly, we construct partial difference sets with Latin square type parameters in the same groups for all $k$ when $\ell$ is even and for all $k<\ell$ when $\ell$ is odd. These constructions provide the first examp…
A zero-sum sequence of integers is a sequence of nonzero terms that sum to 0. Let $k>0$ be an integer and let $[-k,k]$ denote the set of all nonzero integers between $-k$ and $k$. Let $\ell(k)$ be the smallest integer $\ell$ such that any zero-sum sequence with elements from $[-k,k]$ and length greater than $\ell$ contains a proper nonempty zero-sum subsequence. In this paper, we prove a more general result which implies that $\ell(k)=2k-1$ for $k>1$.
In 1990, Spieß\, gave some identities of harmonic numbers including the types of $\sum_{\ell=1}^n\ell^k H_\ell$, $\sum_{\ell=1}^n\ell^k H_{n-\ell}$ and $\sum_{\ell=1}^n\ell^k H_\ell H_{n-\ell}$. In this paper, we derive several formulas of hyperharmonic numbers including $\sum_{\ell=0}^{n} {\ell}^{p} h_{\ell}^{(r)} h_{n-\ell}^{(s)}$ and $\sum_{\ell=0}^n \ell^{p}(h_{\ell}^{(r)})^{2}$. Some more formulas of generalized hyperharmonic numbers are also shown.
V. Bergelson and N. Hindman proved that $IP^{*}$ sets contain all possible finite sum and product of a sum subsystem of any sequence in $\mathbb{N}$. In this article, we will prove this result using Nonstandard analysis.
Kusner asked if $n+1$ points is the maximum number of points in $\mathbb{R}^n$ such that the $\ell_p$ distance $(1<p<\infty)$ between any two points is $1$. We present an improvement to the best known upper bound when $p$ is large in terms of $n$, as well as a generalization of the bound to $s$-distance sets. We also study equilateral sets in the $\ell_p$ sums of Euclidean spaces, deriving upper bounds on the size of an equilateral set for when $p=\infty$, $p$ is even, and for any $1\le p<\infty$.
The set of all $\ell$-zero-sumfree subsets of $\mathbb{Z}/n\mathbb{Z}$ is a simplicial complex denoted by $Δ_{n,\ell}$ We create an algorithm via defining a set of integer partitions we call $(n,\ell)$-congruent partitions in order to compute this complex for moderately-sized parameters $n$ and $\ell$. We also theoretically determine $Δ_{n,\ell}$ for several infinite families of parameters, and compute the intersection lattices and the characteristic polynomials of the corresponding coordinate subspace arrangements.
For fixed $k$ we prove exponential lower bounds on the equilateral number of subspaces of $\ell_{\infty}^n$ of codimension $k$. In particular, we show that if the unit ball of a normed space of dimension $n$ is a centrally symmetric polytope with at most $\frac{4n}{3}-o(n)$ pairs of facets, then it has an equilateral set of cardinality at least $n+1$. These include subspaces of codimension $2$ of $\ell_{\infty}^{n+2}$ for $n\geq 9$ and of codimension $3$ of $\ell_{\infty}^{n+3}$ for $n\geq 15$.
A subset $A$ of a finite abelian group is called $(k,\ell)$-sum-free if $kA \cap \ell A=\emptyset.$ In this paper, we extend this concept to compact abelian groups and study the question of how large a measurable $(k,\ell)$-sum-free set can be. For integers $1 \leq k <\ell$ and a compact abelian group $G$, let $$λ_{k,\ell}(G)=\sup\{ μ(A): kA \cap \ell A =\emptyset \}$$ be the maximum possible size of a $(k,\ell)$-sum-free subset of $G$. We prove that if $G=\mathbb{I} \times M$, where $\mathbb{I}$ is the identity component of $G$, then $$λ_{k, \ell}(G)=\max \left\{ λ_{k, \ell}(M), λ_{k, \ell}(\…
We study extremal problems about sets of integers that do not contain sumsets with summands of prescribed size. We analyse both finite sets and infinite sequences. We also study the connections of these problems with extremal problems of graphs and hypergraphs.
We study arithmetic progressions $\{a,a+b,a+2b,\dots,a+(\ell-1) b\}$, with $\ell\ge 3$, in random subsets of the initial segment of natural numbers $[n]:=\{1,2,\dots, n\}$. Given $p\in[0,1]$ we denote by $[n]_p$ the random subset of $[n]$ which includes every number with probability $p$, independently of one another. The focus lies on sparse random subsets, i.e.\ when $p=p(n)=o(1)$ as $n\to+\infty$. Let $X_\ell$ denote the number of distinct arithmetic progressions of length $\ell$ which are contained in $[n]_p$. We determine the limiting distribution for $X_\ell$ not only for fixed $\ell\ge 3…
We improve a result of Solymosi on sum-products in R, namely, we prove that max{|A+A|,|AA|}\gg |A|^{4/3+c}, where c>0 is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved effectively for sets A from R with |AA| \le |A|^{4/3}.
A graph $G$ is called a sum graph if there is a so-called sum labeling of $G$, i.e. an injective function $\ell: V(G) \rightarrow \mathbb{N}$ such that for every $u,v\in V(G)$ it holds that $uv\in E(G)$ if and only if there exists a vertex $w\in V(G)$ such that $\ell(u)+\ell(v) = \ell(w)$. We say that sum labeling $\ell$ is minimal if there is a vertex $u\in V(G)$ such that $\ell(u)=1$. In this paper, we show that if we relax the conditions (either allow non-injective labelings or consider graphs with loops) then there are sum graphs without a minimal labeling, which partially answers the ques…
Let $\mathscr{M}_{(2,1)}(N)$ be the infimum of the largest sum-free subset of any set of $N$ positive integers. An old conjecture in additive combinatorics asserts that there is a constant $c=c(2,1)$ and a function $ω(N)\to\infty$ as $N\to\infty$, such that $cN+ω(N)<\mathscr{M}_{(2,1)}(N)<(c+o(1))N$. The constant $c(2,1)$ is determined by Eberhard, Green, and Manners, while the existence of $ω(N)$ is still wide open. In this paper, we study the analogous conjecture on $(k,\ell)$-sum-free sets and restricted $(k,\ell)$-sum-free sets. We determine the constant $c(k,\ell)$ for every $(k,\ell)$-su…
Cameron introduced a bijection between the set of sum-free sets and the set of all zero-one sequences. In this paper, we study the sum-free sets of natural numbers corresponding to certain zero-one sequences which contain the Cantor-like sequences and some substitution sequences, etc. Those sum- free sets considered as integer sequences are 2-regular. We also prove that sequences corresponding to certain sum-free sets are automatic.
Let $E(k, \ell)$ denote the smallest integer such that any set of at least $E(k, \ell)$ points in the plane, no three on a line, contains either an empty convex polygon with $k$ vertices or an empty pseudo-triangle with $\ell$ vertices. The existence of $E(k, \ell)$ for positive integers $k, \ell\geq 3$, is the consequence of a result proved by Valtr [Discrete and Computational Geometry, Vol. 37, 565--576, 2007]. In this paper, following a series of new results about the existence of empty pseudo-triangles in point sets with triangular convex hulls, we determine the exact values of $E(k, 5)$ a…
For $\ell\geq 2$ and $h\in \mathbb{Z}[x_1,\dots,x_{\ell}]$ of degree $k\geq 2$, we show that every set $A\subseteq \{1,2,\dots,N\}$ lacking nonzero differences in $h(\mathbb{Z}^{\ell})$ satisfies $|A|\ll_h Ne^{-c(\log N)^μ}$, where $c=c(h)>0$, $μ=[(k-1)^2+1]^{-1}$ if $\ell=2$, and $μ=1/2$ if $\ell\geq 3$, provided $h(\mathbb{Z}^{\ell})$ contains a multiple of every natural number and $h$ satisfies certain nonsingularity conditions. We also explore these conditions in detail, drawing on a variety of tools from algebraic geometry.
Using a slight modification of an argument of Croot, Ruzsa and Schoen we establish a quantitative result on the existence of a dilated copy of any given configuration of integer points in sparse difference sets. More precisely, given any configuration $\{v_1,...,v_\ell\}$ of vectors in $\mathbb{Z}^d$, we show that if $A\subset[1,N]^d$ with $|A|/N^d\geq C N^{-1/\ell}$, then there necessarily exists $r\ne0$ such that $\{rv_1, ...,rv_\ell\}\subseteq A-A$.
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.
