Differencing Methods for Korobov-type exponential sums

We study exponential sums of the form ${\sum }_{n=1}^N{e}^{2\pi ia{b}^n\mathrm{/}m}$ for non-zero integers $a,b,m$. Classically, non-trivial bounds were known for $N\ge \sqrt{m}$ 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 $\exp (\log m\mathrm{/}{\log }_2\log m)\le N$. We include applications to the digits of rational numbers and constructions of normal numbers.