There is an exercise in a paper by George Shakan

George Shakan, Discrete Fourier Transform

say that

Let $A \subset \mathbb{Z}/q\mathbb{Z}$ be any set not containing zero with $|A|>\sqrt2q^{5/8}$. Show that: $$(A+A).(A+A)+A.A+A.A = \mathbb{Z}/q\mathbb{Z}$$ with $$A+B=\{a+b: a\in A, b \in B\}$$ and $$A.B=\{ab: a\in A, b \in B\}$$

so, is there any more ways to expand $\mathbb{Z}/q\mathbb{Z}$? And if there is, what key word I can use to find papers of them?

Thanks.

There is an exercise in a paper by George Shakan

George Shakan, Discrete Fourier Transform

say that

Let $A \subset \mathbb{Z}/q\mathbb{Z}$ be any set not containing zero with $|A|>\sqrt2q^{5/8}$. Show that: $$(A+A).(A+A)+A.A+A.A = \mathbb{Z}/q\mathbb{Z}$$ with $$A+B=\{a+b: a\in A, b \in B\}$$ and $$A.B=\{ab: a\in A, b \in B\}$$

so, is there any more ways to expand $\mathbb{Z}/q\mathbb{Z}$? And if there is, which key word I can use to find papers of them?

Thanks.