Double exponential sums and congruences with intervals and exponential functions modulo a prime

Abstract

Let $p$ be a large prime number and $g$ be any integer of multiplicative order $T$ modulo $p$. We obtain a new estimate of the double exponential sum $$S=\sum _{n\in \mathcal{N}}\left|\sum _{m\in \mathcal{M}}{e}_p(an{g}^m)\right|,\gcd (a,p)=1,$$ where $\mathcal{N}$ and $\mathcal{M}$ are intervals of consecutive integers with $|\mathcal{N}|=N$ and $|\mathcal{M}|=M<T$ elements. One representative example is the following consequence of the main result: if $N=M\approx p^{1/3}$, then $|S|<N^{2-1/8+o(1)}$. We then apply our estimate to obtain new results on additive congruences involving intervals and exponential functions.

References in corpus (1)

• Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications