Th main result of this preprint of Imre Ruzsa implies the following
Corollary (Ruzsa): For every $r\in\mathbb N$ there exists a real number $\alpha<1$ and a positive integer $m$ such that for every integer $q\ge m$ the cyclic group $C_q$ of order $q$ contains a subset $A$ such that $A-A=C_q$ but the sum-set $A^{+r}:=\{x_1+\dots+x_r:x_1,\dots,x_r\in A\}$ has cardinality $|A^{+r}|<q^\alpha<|C_q|$.
On the other hand, Nathanson citing the result of Ruzsa mentions only a weaker version of the above corollary, which was proved earlier by Haight:
Theorem (Haight): For every $r\in\mathbb N$ there exists a finite cyclic group $G$ containing a subset $A\subset G$ such that $A-A=G$ and $A^{+r}\ne G$.
At the end of his preprint, Nathanson asks the following
Problem (Nathanson): Let $r$ be a positive integer and $\varepsilon>0$. Is there a cyclic group $G$ of prime order and a set $A\subset G$ such that $A-A=G$ and $|A^{+r}|<\varepsilon |G|$? Do such sets exist for all cyclic group of sufficiently large prime cardinality?
It seems that the mentioned Corollary of Ruzsa answers this question of Nathanson. So, why then Nathanson asked this question? And why the preprint of Ruzsa (of 2016) is still unpublished? I would like to apply it for constructing some pathological subsemigroups in products of cyclic groups.
For me it would be sufficient to have the following weaker version of Ruzsa's Corollary:
Theorem (or still a Conjecture?): For every $r\in\mathbb N$ there exists a positive integer $m$ such that for every integer $q\ge m$ the cyclic group $C_q$ of order $q$ contains a subset $A$ such that $A-A=C_q$ but $A^{+r}\ne C_q$.
So, is the above theorem true and can I refer to the paper of Ruzsa for its proof? Or there is a gap somewhere? Please help!
Added in Edit. Theorem 1 of this paper of Haight shows that the last Theorem is true (so is not a conjecture anymore) and the answer to the question of Nathanson also is affirmative.
