Differencing methods for Korobov-type exponential sums

Abstract

We study exponential sums of the form $$\Sigma_{n=1}^N\;e^{{2\pi}iab^{n}/m}$$ for nonzero integers a, b, m. Classically, non-trivial bounds were known for N ≥ √m by Korobov, and this range has been extended significantly by Bourgain as a result of his and others’ work on the sum-product phenomenon. Let P be a finite set of primes and let m be a large integer whose primes factors all belong to P. We use a variant of the Weyl-van der Corput method of differencing to give more explicit bounds that become non-trivial around the time when exp(log m/ log2 logm) ≤ N. We include applications to the digits of rational numbers and constructions of normal numbers.