19 problems

Let $G$ be a finite abelian group of order $n$, let $p$ be the smallest prime divisor of $n$, and let $\rho\hat{\;} (G,m,h)$ denote the minimum restricted $h$-fold sumset size amon…

Gallardo–Grekos–Pihko conjecture. There is a constant $c$ such that, whenever

Small-set restricted-sumset conjecture. If

Bounded-degree restricted-sumset conjecture. If the maximum degree of $\mathcal{R}$ on $B$ is at most $\Delta$, then

Lev's conjecture.

Lev–Freiman conjecture. One has

Let $q$ be an odd square with $q\geq 49$, and let $A\subset\mathbb{F}_q$ satisfy … Here $A\mathbin{\hat{+}}A$ is the restricted sumset of $A$, and $S_2(\mathbb{F}_q)$ is the subgro…

Let $G$ be an abelian group, and let $p$ be the smallest prime divisor of the order of $G$. For positive integers $m$ and $h$ with $m\leq p$, write $\rho\hat{\;} (G,m,h)$ for the m…

Let $G$ be an additive group with $|G|>1$. For finite subsets $A_1,\ldots,A_n$ of $G$, define … and … Let $n>1$ and assume that $|A_i|>1$ for every $i=1,\ldots,n$. The Sun restrict…

Let $G$ be a finite group with $|G|>1$, and let $p(G)$ be the minimum of the orders of the nonzero elements of $G$, with $p(G)=+\infty$ if every nonzero element has infinite order.…

Let $n$ be a positive integer and let $m\leq n$. For an $m$-subset $A$ of $\mathbb{Z}_n$, write $\Sigma A=\bigcup_{h=0}^{\infty}h\hat{\;}A$ for its restricted sumset over all numbe…

Let $G$ be a finite abelian group of order $n$, let $p$ be the smallest prime divisor of $n$, and let $A$ be an $m$-subset of $G$. Inverse restricted-sumset conjecture. If…

Let $G$ be a finite abelian group, and let $\rho(G,m,2)$ and $\rho\hat{\;} (G,m,2)$ denote respectively the minimum unrestricted and restricted double sumset sizes among $m$-subset…

Let $\mathbb{Z}_n$ be the cyclic group of order $n$. For an $m$-subset of $\mathbb{Z}_n$, let $\rho\hat{\;} (\mathbb{Z}_n,m,h)$ denote the minimum size of its restricted $h$-fold s…

Let $G$ be a finite abelian group of odd order $n$, and let $m$ be exceptional for $G$ when, for every finite nontrivial minimal subgroup $M$ of $G$, one has $m\equiv1\pmod{|M|}$.…

Let $G$ be a finite abelian group, let $m\leq |G|$, and write $\rho\hat{\;} (G,m,2)$ and $\rho(G,m,2)$ for the minimum sizes of restricted and unrestricted 2-fold sumsets of an $m$…

For positive integers $n$ and $m$ with $3\leq m\leq n$, let $\rho\hat{\;} (\mathbb{Z}_n,m,2)$ be the minimum size of a restricted 2-fold sumset of an $m$-subset of $\mathbb{Z}_n$,…

Let $A$ be a subset of a finite group, and let $A\dot{+}A$ denote the sumset formed using distinct elements of $A$. Erdős–Heilbronn conjecture. If the addition is restricted to dis…

Let $p$ be a prime, let $A$ and $B$ be non-empty subsets of $\mathbb Z_p$, and let $S\subseteq\mathbb Z_p$. Define … Theorem 2 gives $|C|\geqslant\min\{p,\,|A|+|B|-2|S|-1\}$. Heilb…