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Register a new userSets with few differences in abelian groups
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Mitchell Lee
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Let $(G, +)$ be an abelian group. In 2004, Eliahou and Kervaire found an explicit formula for the smallest possible cardinality of the sumset $A+A$, where $A subseteq G$ has fixed cardinality $r$. We consider instead the smallest possible cardinality of the difference set $A-A$, which is always greater than or equal to the smallest possible cardinality of $A+A$ and can be strictly greater. We conjecture a formula for this quantity and prove the conjecture in the case that $G$ is a cyclic group or a vector space over a finite field. This resolves a conjecture of Bajnok and Matzke on signed sumsets.
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We show that, in contrast to the integers setting, almost all even order abelian groups $G$ have exponentially fewer maximal sum-free sets than $2^{mu(G)/2}$, where $mu(G)$ denotes the size of a largest sum-free set in $G$. This confirms a conjecture of Balogh, Liu, Sharifzadeh and Treglown.
The known families of difference sets can be subdivided into three classes: difference sets with Singer parameters, cyclotomic difference sets, and difference sets with gcd$(v,n)>1$. It is remarkable that all the known difference sets with gcd$(v,n)>1$ have the so-called character divisibility property. In 1997, Jungnickel and Schmidt posed the problem of constructing difference sets with gcd$(v,n)>1$ that do not satisfy this property. In an attempt to attack this problem, we use difference sets with three nontrivial character values as candidates, and get some necessary conditions.
Let $v$ be an odd real polynomial (i.e. a polynomial of the form $sum_{j=1}^ell a_jx^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $mathcal R(v,epsilon)={ninmathbb{N},|,|v(n)|{<epsilon}}$ where $|cdot|$ denotes the distance to the closest integer. We then apply the new diophantine results to obtain applications to ergodic theory and combinatorics. In particular, we obtain a new characterization of weakly mixing systems as well as a new variant of Furstenberg-Sarkozy theorem.
We show that any connected Cayley graph $Gamma$ on an Abelian group of order $2n$ and degree $tilde{Omega}(log n)$ has at most $2^{n+1}(1 + o(1))$ independent sets. This bound is tight up to to the $o(1)$ term when $Gamma$ is bipartite. Our proof is based on Sapozhenkos graph container method and uses the Pl{u}nnecke-Rusza-Petridis inequality from additive combinatorics.
A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $gin G$ can be written as the difference $g=ab^{-1}$ of some elements $a,bin B$. The smallest cardinality $|B|$ of a difference basis $Bsubset G$ is called the difference size of $G$ and is denoted by $Delta[G]$. The fraction $eth[G]:=frac{Delta[G]}{sqrt{|G|}}$ is called the difference characteristic of $G$. Using properies of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite Abelian groups. In particular, we prove that for a prime number $pge 11$, any finite Abelian $p$-group $G$ has difference characteristic $eth[G]<frac{sqrt{p}-1}{sqrt{p}-3}cdotsup_{kinmathbb N}eth[C_{p^k}]<sqrt{2}cdotfrac{sqrt{p}-1}{sqrt{p}-3}$. Also we calculate the difference sizes of all Abelian groups of cardinality $<96$.
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