Distribution of exponential functions with squarefull exponent in residue rings.
Let \(\mathcal{S}\) a set of positive integers. For a real \(x\geq 1\) we put \({\mathcal{S}[x]=\mathcal{S}\cap[1,x]}\) and \(T_a(m,x)=\sum_{n\in \mathcal{S}[x]} \exp(2ia\pi\theta^n/m)\) where \(\theta\) is a fixed integer with \(gcd(\theta, m)=1\). The author and others studied the above exponential sum in different cases. For example one can see [\textit{W. D. Banks, A. Conflitti, J. B. Friedlander} and \textit{I. E. Shparlinski}, Compos. Math. 140, No. 1, 15--30 (2004; Zbl 1060.11045)] when \(\mathcal{S}\) is the set of square free number or [\textit{J. B. Friedlander, J. Hansen} and \textit{I. E. Shparlinski}, Mathematika 47, No. 1--2, 75--85 (2000; Zbl 1017.11045)] if \(\mathcal{S}\) is the set of square or hight powers. In this paper the author continues the investigation in this direction and he studies the exponential sum \(T_a(m,x)\) in the case where \(\mathcal{S}\) is the set of squarefull integers. If we put \(t\) the order of \(\theta\) in \(\mathbb{Z}^*_m\), the group of units of the ring \( \mathbb{Z}_m\) , \(|\mathcal{S}[x]| =\sum_{n\in \mathcal{S}[x]}1\) and \(\varepsilon\) a fixed positif real number, he proves that \newline \(\bullet\) For a prime \(m=p\) \[ \max_{\gcd(a,p)=1}\mid T_a(p,x)\mid \ll |\mathcal{S}[x]| (x^{-1/6}t^{1/3}+t^{-1/4}p^{1/8})p^{\varepsilon} \] \(\bullet\) For any \(m\) and any positive divisor \(D\) of \(m\), \[ \max_{\gcd(a,m)=D}\mid T_a(m,x)\mid\ll |\mathcal{S}[x]| (x^{-1/6}t^{1/3}+D^{1/8}t^{-11/32}m^{5/16})m^{\varepsilon}. \] According to the well-known Erdős-Turán relation between discrepancy and exponential sums, he deduces the discrepancy of the sequence of fraction parts \(\{\frac{\theta^n}{m}\}\), \(n\in \mathcal{S}[x]\).
scientific article; zbMATH DE number 2124644
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