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.
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The author acknowledges assistance from the Research and Training Group grant DMS-1344994 funded by the National Science Foundation.
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Vandehey, J. Differencing methods for Korobov-type exponential sums. JAMA 138, 405–439 (2019). https://doi.org/10.1007/s11854-019-0038-2
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DOI: https://doi.org/10.1007/s11854-019-0038-2