arXiv subjects

Bela Bajnok

Publications and source records attributed to Bela Bajnok.

23 records · Page 2Linked to original sources

arXiv2016

On Asymptotic Approximate Groups of Integers

Let $r$ be a positive integer, and let $A$ be a nonempty finite set of at least two integers. We let $\tilde{C}_r(A)$ denote the {\em asymptotic $r$-covering number} of $A$, that is, the smallest integer value of $l$ for which, for all sufficiently large positive integers $h$, the $rh$-fold sumset of $A$ is contained in at most $l$ translates of the $h$-fold sumset of $A$. Nathanson proved that $\tilde{C}_r(A)$ is always at most $r+1$; here we extend this result to prove that $\tilde{C}_r(A)$ is always at least $r$, and determine all sets $A$ for which $\tilde{C}_r(A)=r$.

math.NT↗

arXiv2014

The $h$-critical number of finite abelian groups

For a finite abelian group $G$ and a positive integer $h$, the unrestricted (resp.~restricted) $h$-critical number $χ(G,h)$ (resp.~$χ\hat{\;}(G,h)$) of $G$ is defined to be the minimum value of $m$, if exists, for which the $h$-fold unrestricted (resp.~restricted) sumset of every $m$-subset of $G$ equals $G$ itself. Here we determine $χ(G,h)$ for all $G$ and $h$; and prove several results for $χ\hat{\;}(G,h)$, including the cases of any $G$ and $h = 2$, any $G$ and large $h$, and any $h$ for the cyclic group $\mathbb{Z}_n$ of even order. We also provide a lower bound for $χ\hat{\;}(\mathbb{Z}_n,3)$ that we believe is exact for every $n$---this conjecture is a generalization of the one made by Gallardo, Grekos, et al.~that was proved (for large $n$) by Lev.

math.NT↗

arXiv2014

The Minimum Size of Signed Sumsets

For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$ρ(G, m, h) = \min \{|hA| \; : \; A \subseteq G, |A|=m\}$$ and $$ρ_{\pm} (G, m, h) = \min \{|h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and $h_{\pm} A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $ρ(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. Here we prove that $ρ_{\pm}(G, m, h)$ equals $ρ(G, m, h)$ when $G$ is cyclic, and establish an upper bound for $ρ_{\pm} (G, m, h)$ that we believe gives the exact value for all $G$, $m$, and $h$.

math.NT↗

arXiv2014

On the Minimum Size of Signed Sumsets in Elementary Abelian Groups

For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$ρ(G, m, h) = \min \{|hA| \; : \; A \subseteq G, |A|=m\}$$ and $$ρ_{\pm} (G, m, h) = \min \{|h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and $h_{\pm} A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $ρ(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. In previous work we provided an upper bound for $ρ_{\pm} (G, m, h)$ that we believe is exact, and proved that $ρ_{\pm} (G, m, h)$ agrees with $ρ(G, m, h)$ when $G$ is cyclic. Here we study $ρ_{\pm} (G, m, h)$ for elementary abelian groups $G$; in particular, we determine all values of $m$ for which $ρ_{\pm} (\mathbb{Z}_p^2, m, 2)$ equals $ρ(\mathbb{Z}_p^2, m, 2)$ for a given prime $p$.

math.NT↗

arXiv2008

On the maximum size of a $(k,l)$-sum-free subset of an abelian group

A subset $A$ of a given finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ does not equal the sum of $l$ (not necessarily distinct) elements of $A$. We are interested in finding the maximum size $λ_{k,l}(G)$ of a $(k,l)$-sum-free subset in $G$. A $(2,1)$-sum-free set is simply called a sum-free set. The maximum size of a sum-free set in the cyclic group $\mathbb{Z}_n$ was found almost forty years ago by Diamanda and Yap; the general case for arbitrary finite abelian groups was recently settled by Green and Ruzsa. Here we find the value of $λ_{3,1}(\mathbb{Z}_n)$. More generally, a recent paper of Hamidoune and Plagne examines $(k,l)$-sum-free sets in $G$ when $k-l$ and the order of $G$ are relatively prime; we extend their results to see what happens without this assumption.

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