Zeros and local extrema of trigonometric sums (Q2487079)

An exponential sum of the type \[ \sum_{a<x\leq b}h(x)\cos (2\pi (\alpha _1x +\dotsb + \alpha _n x^n)), \] or a similar sum involving the sine function, is a periodic function with period 1 in the coefficients \(\alpha _i\). In particular, if we fix all coefficients except \(\alpha _m =\alpha \), we get a periodic function of \(\alpha \). One of the topics in this paper is the distribution of the zeros of this function if the weight function \(h(x)\) is assumed to take values \(0\) or \(\pm 1\), not always 0. As a corollary of a general theorem, it is proved that if \(m\geq 2\), \(a \geq 3\), \(b \geq 2^m\), and \(2a \geq b\), then the number of zeros in the interval \([u,v]\subseteq [0,1]\) is at least \((v-u)(8\ln b)^{-1}b^{m-1}-1\). Thus if \(v-u \geq 8b^{-m+1}\ln b\), then there is a zero in the interval \([u,v]\). Several other analogous problems are discussed as well.