Small Sets with Large Difference Sets

For every $\epsilon > 0$ and $k \in \mathbb{N}$, Haight constructed a set $A \subset \mathbb{Z}_N$ ($\mathbb{Z}_N$ stands for the integers modulo $N$) for a suitable $N$, such that $A-A = \mathbb{Z}_N$ and $|kA| < \epsilon N$. Recently, Nathanson posed the problem of constructing sets $A \subset \mathbb{Z}_N$ for given polynomials $p$ and $q$, such that $p(A) = \mathbb{Z}_N$ and $|q(A)| < \epsilon N$, where $p(A)$ is the set $\{p(a_1, a_2, \dots, a_n)\phantom{.}\colon\phantom{.}a_1, a_2, \dots, a_n \in A\}$, when $p$ has $n$ variables. In this paper, we give a partial answer to Nathanson's question. For every $k \in \mathbb{N}$ and $\epsilon > 0$, we find a set $A \subset \mathbb{Z}_N$ for suitable $N$, such that $A- A = \mathbb{Z}_N$, but $|A^2 + kA| < \epsilon N$, where $A^2 + kA = \{a_1a_2 + b_1 + b_2 + \dots + b_k\phantom{.}\colon\phantom{.}a_1, a_2,b_1, \dots, b_k \in A\}$. We also extend this result to construct, for every $k \in \mathbb{N}$ and $\epsilon > 0$, a set $A \subset \mathbb{Z}_N$ for suitable $N$, such that $A- A = \mathbb{Z}_N$, but $|3A^2 + kA| < \epsilon N$, where $3A^2 + kA = \{a_1a_2 + a_3a_4 + a_5a_6 + b_1 + b_2 + \dots + b_k\phantom{.}\colon\phantom{.}a_1, \dots, a_6,b_1, \dots, b_k \in A\}$.