Exponential sums and arithmetic functions running \(\\\) on subintervals

Abstract. Let \({\cal M}_1\) stand for the set of all complex valued multiplicative functions satisfying \(|f(n)|\le 1\) for all \(n\in {\mathbb N}\). Some fifty years ago, Hedi Daboussi proved that, given any irrational number \(\alpha\), we have \(\sup_{f\in {\cal M}_1} \sum_{n\le x} f(n) e^{2\pi i \alpha n} =o(x)\) as \(x\to \infty\). We show that the same result holds if \(f\) satisfies the additional condition \(|f(n)|=1\) for all \(n\in {\mathbb N}\) and if we replace the exponential expression \(e^{2\pi i \alpha n}\) by \(h(u(n))\), where \(h\) is the characteristic function of a set made of a collection of subintervals of \([0,1)\) and \(u(n)\) is an arithmetic function with specific properties.

Key words and phrases. Exponential sums, multiplicative functions.