An average bound for character sums with some counter-dependent recurrence sequences
There are several recent papers in the literature giving nontrivial upper bounds for multiplicative and additive character sums with \(n!\) and other related combinatorial sequences. Inspired by these papers, the authors take a prime \(p\) and the sequence \((a_{\alpha}(n))_{n\geq 0}\) satisfying the recurrence \(a_{\alpha}(n)=na_{\alpha}(n-1)+1\) for all \(n\geq 1\) and with \(a_{\alpha}(0)=\alpha\in {\mathbb F}_p\), the finite field with \(p\) elements. Note that when \(\alpha=0\), the general term of the above sequence is \(a_0(n)=\lfloor (e-2)n!\rfloor\). The paper deals with the sum \[ W_{\nu}(\chi,\psi,N)={{1}\over {p}}\sum_{\alpha=0}^{p-1} \left|\sum_{n=0}^{N-1} \chi(a_{\alpha}(n))\psi(a_{\alpha}(n))\right|^{\nu}, \] where \((\chi,\psi)\) is a pair consisting of a multiplicative and an additive character of \({\mathbb F}_p\), not both trivial. Their main result is that if \(\nu\) is a nonnegative integer and \(N\leq p\), then \[ W_{2\mu}(\chi,\psi,N)=O\left(N^{2\mu-1} \max\left\{Np^{-1/2},N^{1/(\mu+1)}\right\}\right). \] The proof starts with the inner sum for \(W_1(\chi,\psi,N)\) and observes that its size does not change much is we shift the range of summation from \([0,N-1]\) to \([1,N]\), or \([2,N+1]\), or, in general, to \([K,N+K-1]\), where \(K\) is some suitable chosen parameter. With this observation, one can replace the inner sum for \(W_1(\chi,\psi,N)\) by a double sum inside which one can reverse the order of summation. Now the inner sums of \(W_{2\mu}(\chi,\psi,N)\) are related to the above double sums via Hölder's inequality. For the rest of the proof, one reduces the problem to an instance where one can use Weil's estimate for character sums with nonconstant rational functions in \({\mathbb F}_p\). A nontrivial amount of effort is spend insuring that the various rational functions that show up and to which Weil's bound is applied are indeed nonconstant. The paper concludes by observing that their result implies via the Cauchy-Schwartz inequality a nontrivial upper bound for \(W_{2\mu-1}(\chi,\psi,N)\). In particular, \(W_1(\chi,\psi,N)\ll N^{3/4}\).
scientific article; zbMATH DE number 5614516
scientific article