Differencing Methods for Korobov-type exponential sums

2016/06/25 by Vandehey, Joseph
#11A63 #11K16 #11L07 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1606.07911

Abstract

We study exponential sums of the form ∑_n=1^N e^{2πi a bn/m} for non-zero 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. We use a new technique, similar to the Weyl-van der Corput method of differencing, to give more explicit bounds bounds that become non-trivial around the time when exp(log m/log₂log m) ≤ N. We include applications to the digits of rational numbers and constructions of normal numbers.

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