We are interested in a sum-product type estimate. Let $p$ be an odd prime, and let $A$ be the order $p$ subgroup of $(\mathbb{Z}/p^2\mathbb{Z})^\times$. That is, let $A = \langle g^p \rangle$, where $(\mathbb{Z}/p^2\mathbb{Z})^\times = \langle g \rangle$. We seek an estimate of the size of the sum set $$|A + A|.$$ The standard references, e.g. Additive Combinatorics by Tao & Vu, do not discuss this problem in the setting of $\mathbb{Z}/p^2\mathbb{Z}$, but instead focus on analogous results in finite fields. Similar estimates address situations where $|A + A|$ is small (e.g., on the order of $K|A|$), whereas in our situation, the product set $|A\cdot A|$ is very small, and we wish to show that $|A+A|$ must be large. Do results exist that address this problem more directly?

We are interested in a sum-product type estimate. Let $p$ be an odd prime, and let $A$ be the order $p-1$ subgroup of $(\mathbb{Z}/p^2\mathbb{Z})^\times$. That is, let $A = \langle g^p \rangle$, where $(\mathbb{Z}/p^2\mathbb{Z})^\times = \langle g \rangle$. We seek an estimate of the size of the sum set $$|A + A|.$$ The standard references, e.g. Additive Combinatorics by Tao & Vu, do not discuss this problem in the setting of $\mathbb{Z}/p^2\mathbb{Z}$, but instead focus on analogous results in finite fields. Similar estimates address situations where $|A + A|$ is small (e.g., on the order of $K|A|$), whereas in our situation, the product set $|A\cdot A|$ is very small, and we wish to show that $|A+A|$ must be large. Do results exist that address this problem more directly?