Cancellation in a short exponential sum
Let \(q\) be an odd integer, \(\tau\) be the order of 2 modulo \(q\) and let \(\xi\) be a primitive \(q\)th root of unity. In this paper bounds for the sum \[ s(\xi)=\sum\limits_{r=1}^{\tau}\xi^{2^r} \] are obtained. The behavior of \(s(\xi)\) when \(q\) diverges along a sequence for which the quotient \(\tau/\mathcal{L}\) belongs to the range \([\mu,\,\nu]\) with \(1\leq \mu <\nu\), \(\nu-1\) is small enough and \(\mathcal{L}=[\log_2q]\) is studied. If \(U(l,\,k)\) is the number of representation of \(l\) as sum of \(k\) powers of two and \(\varrho\) is a constant such that \(\max\limits_{l}\{U(l,\,k)\}\leq (\varrho k)^k.k^{o(1)}\) holds for enough large \(k\), the upper bound \[ \max\limits_{\xi :\xi ^q=1\atop{\xi \,\text{primitive}}} |s(\xi)|\leq \tau-c_{\mu,\nu}(\varrho)+o_{\varrho,\mu,\nu}(1), \] where \[ c_{\mu,\nu}(\varrho)=\max\limits_{x\geq 1}\bigg\{-\frac{\nu-1/2}{\mu}x^2\log 2+x\log(x/\varrho) \bigg\} \] is obtained. An explicit value for \(\varrho\) is obtained and thereby some improvements are mentioned.
scientific article; zbMATH DE number 5759069