Multiplicative character sums and products of sparse integers in residue classes
Let \( n=\sum_{j\geq0}a_{j}(n)2^{j} \), with \( a_{j}(n)=0, 1 \) be the binary representation of \(n\) and let \( \mathcal{G}_{s}(r) \) be the set of integers with \(r\) binary integers such that the sum of the digits is equal to \(s\). The authors of this paper discuss character sums of the form \(S_{s}(r,\chi,f)\), where \(\chi\) is a non-principal multiplicative character of the finite field \( \mathbb{R}_{p} \) with \(p\) elements, and \(f(X)\) is a polynomial in \( \mathbb{R}_{p} \). They prove three theorems. One of these is: Theorem 2. For any \( \gamma > \rho_{0}=0.11002786... \), there exists some \( \eta > 0 \) such that if \( p>2^{r}=p^{1+o(1)} \) and \(r/2\geq s\geq \gamma r\), then for any nontrivial multiplicative character \(\chi\) modulo \(p\), we have \(S_{s}(r,\chi, f) \ll {r \choose s}^{1-\eta}\).
scientific article; zbMATH DE number 6083230