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Imre Z. Ruzsa

Publications and source records attributed to Imre Z. Ruzsa.

At least 19 recordsLinked to original sources
arXiv2025

Densitometria I. Discrete groups

An upper mean here is a subadditive functional $\overline M$ defined on bounded functions on a commutative group which has, beside some natural requirements, the property we call restricted additivity: if $g(x)= f(x)+f(x+t)$, then $\overline{M} (g)= 2 \overline{M} (f)$. This tries to grasp that it should not depend on local properties. This naturally induces a lower mean, and when they coincide it is the mean. Restriction to 0--1 valued functions (sets) is a density. We answer the following questions: Given a functional defined on a subset of all functions, when is it a mean? Given a functional, which is a mean, how do we find the upper mean it came from? Is it unique? Given a function $f$, what are the possible values of $\overline M(f)$, for upper means $\overline M$? In particular, we find the extremal means and give several expressions for it. We propose the names ``lowest and uppermost mean'' for them to replace the not really justified names ``lower and upper Banach mean and density''. We also consider analogous questions for densities, with partial answers only.

math.CA↗
arXiv2025

The fractional chromatic number of the plane is at least 4

We prove that the fractional chromatic number $χ_f(\mathbb R^2)$ of the unit distance graph of the Euclidean plane is greater than or equal to $4$. Interestingly, however, we cannot present a finite subgraph $G$ of the plane such that $χ_f(G)\ge 4$. Instead, we utilize the concept of the geometric fractional chromatic number $χ_{gf}(G)$, which was introduced recently in connection with density bounds for 1-avoiding sets. First, as $G$ ranges over finite subgraphs of the plane, we establish that the supremum of $χ_f(G)$ is the same as that of $χ_{gf}(G)$. The proof exploits the amenability of the group of Euclidean transformations in dimension 2 and, as such, we do not know whether the analogous statement holds in higher dimensions. We then present a specific planar unit distance graph $G$ on 27 vertices such that $χ_{gf}(G)=4$, and conclude $χ_f(\mathbb R^2)\ge 4$ as a corollary. As another main result we show that the finitary fractional chromatic number and the Hall ratio of the plane are equal. As a consequence, we conclude that there exist finite unit distance graphs with independence ratio $\frac{1}{4}+\varepsilon$, while we conjecture that the value $\frac{1}{4}$ cannot be reached.

math.CO↗
arXiv2023

Beurling integers with lacunarity

We present examples of multiplicative semigroups of positive reals (Beurling's generalized integers) with gaps bounded from below.

math.NT↗
arXiv2018

Triangulations and a discrete Brunn-Minkowski inequality in the plane

For a set $A$ of points in the plane, not all collinear, we denote by ${\rm tr}(A)$ the number of triangles in any triangulation of $A$; that is, ${\rm tr}(A) = 2i+b-2$ where $b$ and $i$ are the numbers of points of $A$ in the boundary and the interior of $[A]$ (we use $[A]$ to denote "convex hull of $A$"). We conjecture the following analogue of the Brunn-Minkowski inequality: for any two point sets $A,B \subset {\mathbb R}^2$ one has \[ {\rm tr}(A+B)^{\frac12}\geq {\rm tr}(A)^{\frac12}+{\rm tr}(B)^{\frac12}. \] We prove this conjecture in several cases: if $[A]=[B]$, if $B=A\cup\{b\}$, if $|B|=3$, or if none of $A$ or $B$ has interior points.

math.CO↗
arXiv2018

Nearly subadditive sequences

We show that the de Bruijn-Erdős condition for the error term in their improvement of Fekete's Lemma is not only sufficient but also necessary in the following strong sense. Suppose that given a sequence $0\leq f(1)\leq f(2)\leq f(3)\leq \dots $ such that \begin{equation}\sum_{ n=1}^{\infty} f(n)/n^2 = \infty. \end{equation} Then, there exists a sequence $\{b(n)\}_{n=1,2,\dots}$ satisfying \begin{equation}\label{eq1} b(n+m) \leq b(n) + b(m) + f(n+m) \end{equation} such that the sequence of slopes $\{ b(n)/n\}_{n=1,2,\dots}$ takes every rational number. When the series is bounded we improve their result as follows. If there exist $N$ and real $μ>1$ such that near $f$-subadditivity holds for all pairs $(n,m)$ with $N\leq n\leq m \leq μn$, then $\lim_n b(n)/n $ exists.

math.CO↗
arXiv2018

On the size of the set $AA+A$

It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A| \gg |A|^{\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A \subset \mathbb R$ such that $|AA+A|=o(|A|^2)$, disproving a conjecture of Balog.

math.CO↗
arXiv2017

Average Goldbach and the Quasi-Riemann Hypothesis

We prove that a good average order on the Goldbach generating function implies that the real parts of the non-trivial zeros of the Riemann zeta function are strictly less than 1. This together with existing results establishes an equivalence between such asymptotics and the Riemann Hypothesis.

math.NT↗
arXiv2017

Convex sequences may have thin additive bases

For a fixed $c > 0$ we construct an arbitrarily large set $B$ of size $n$ such that its sum set $B+B$ contains a convex sequence of size $cn^2$, answering a question of Hegarty.

math.CO↗
arXiv2016

Monochromatic paths for the integers

Recall that van der Waerden's theorem states that any finite coloring of the naturals has arbitrarily long monochromatic arithmetic sequences. We explore questions about the set of differences of those sequences.

math.CO↗
arXiv2016

More differences than multiple sums

We compare the size of the difference set $A-A$ to that of the set $kA$ of $k$-fold sums. We show the existence of sets such that $|kA| < |A-A|^{a_k}$ with $a_k<1$.

math.NT↗
arXiv2015

Better bounds for planar sets avoiding unit distances

A $1$-avoiding set is a subset of $\mathbb{R}^n$ that does not contain pairs of points at distance $1$. Let $m_1(\mathbb{R}^n)$ denote the maximum fraction of $\mathbb{R}^n$ that can be covered by a measurable $1$-avoiding set. We prove two results. First, we show that any $1$-avoiding set in $\mathbb{R}^n$ ($n\ge 2$) that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than $1$ and points from distinct blocks lie farther than $1$ unit of distance apart from each other) has density strictly less than $1/2^n$. For the special case of sets with block structure this proves a conjecture of Erdős asserting that $m_1(\mathbb{R}^2) < 1/4$. Second, we use linear programming and harmonic analysis to show that $m_1(\mathbb{R}^2) \leq 0.258795$.

math.MG↗
arXiv2015

Exact additive complements

Let $A,B$ be sets of positive integers such that $A+B$ contains all but finitely many positive integers. Sárközy and Szemerédi proved that if $ A(x)B(x)/x \to 1$, then $A(x)B(x)-x \to \infty $. Chen and Fang considerably improved Sárközy and Szemerédi's bound. We further improve their estimate and show by an example that our result is nearly best possible.

math.NT↗
arXiv2015

Carries and the arithmetic progression structure of sets

If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues modulo $m$. When adding two integers with last digits $a_1, a_2 \in A$, we find the unique $a \in A$ such that $a_1 + a_2 \equiv a$ mod $m$, and call $(a_1 + a_2 -a)/m$ the carry. Carries occur also when addition is done modulo $m^2$, with $A$ chosen as a set of coset representatives for the cyclic group $\mathbb{Z}/m \mathbb{Z} \subseteq \mathbb{Z}/m^2\mathbb{Z}$. It is a natural to look for sets $A$ which minimize the number of different carries. In a recent paper, Diaconis, Shao and Soundararajan proved that, when $m=p$, $p$ prime, the only set $A$ which induces two distinct carries, i. e. with $A+A \subseteq \{ x, y \}+A$ for some $x, y \in \mathbb{Z}/p^2\mathbb{Z}$, is the arithmetic progression $[0, p-1]$, up to certain linear transformations. We present a generalization of the result above to the case of generic modulus $m^2$, and show how this is connected to the uniqueness of the representation of sets as a minimal number of arithmetic progression of same difference.

math.NT↗
arXiv2013

Squares and difference sets in finite fields

For infinitely many primes $p=4k+1$ we give a slightly improved upper bound for the maximal cardinality of a set $B\subset \ZZ_p$ such that the difference set $B-B$ contains only quadratic residues. Namely, instead of the "trivial" bound $|B|\leq \sqrt{p}$ we prove $|B|\leq \sqrt{p}-1$, under suitable conditions on $p$. The new bound is valid for approximately three quarters of the primes $p=4k+1$.

math.CO↗
arXiv2012

Difference sets and positive exponential sums I. General properties

We describe general connections between intersective properties of sets in Abelian groups and positive exponential sums. In particular, given a set $A$ the maximal size of a set whose difference set avoids $A$ will be related to positive exponential sums using frequencies from $A$.

math.CO↗