Summary: We prove that if \(A \subset {\mathbb F}_q\) is such that \[ |A|>q^{{1/2}+\frac{1}{2d}}, \] then \[ {\mathbb F}_q^{*} \subset dA^2=\underbrace{A^2+\dots +A^2}_{ d \text{ times}}, \] where \[ A^2=\{a \cdot a': a,a' \in A\} \] and where \({\mathbb F}_q^{*}\) denotes the multiplicative group of the finite field \({\mathbb F}_q\). In particular, we cover \({\mathbb F}_q^{*}\) by \(A^2+A^2\) if \(|A|>q^{{3/4}}\). Furthermore, we prove that if \[ |A| \geq C_{\text{size}}^{\frac{1}{d}}q^{{1/2}+\frac{1}{2(2d-1)}}, \] then \[ |dA^2| \geq q \cdot \frac{C^2_{\text{size}}}{C^2_{\text{size}}+1}. \] Thus \(dA^2\) contains a positive proportion of the elements of \({\mathbb F}_q\) under a considerably weaker size assumption. We use the geometry of \({\mathbb F}_q^d\), averages over hyper-planes and orthogonality properties of character sums. In particular, we see that using operators that are smoothing on \(L^2\) in the Euclidean setting leads to non-trivial arithmetic consequences in the context of finite fields.
For the entire collection see [Zbl 1143.42002].