# $m$-thick sets with small $n$-fold sumsets in finite cyclic groups

Problem. Is it true that for every positive integers $n,m$ there exists a subset $A_{n,m}$ of a finite cyclic group $G$ having the following two properties:

$(\Sigma_n)$ the $n$-fold sum $A_{n,m}^{+n}=\{x_1+\dots+x_n:x_1,\dots,x_n\in A_{n,m}\}$ does not coincide with $G$;

$(T_m$) the set $A_{n,m}$ is $m$-thick in the sense that for any subset $B\subset G$ of cardinality $|B|\le m$ there exists $g\in G$ such that $g+B\subset A_{n,m}$?

Remark 1. The answer to this problem is affirmative for any $n\in\mathbb N$ and $m=2$. The first example of a set $A_{n,2}$ was constructed by Haight in 1973 and then developed by many mathematicians, in particular, Rusza and Nathanson. But what is known about sets $A_{n,m}$ for $m>2$?

Remark 2. The set $\mathbb N$ of positive integer numbers in the infinite cyclic group $\mathbb Z$ trivially has the properties $\Sigma_n$ and $T_m$.