Bounds of Double Exponential Sums: Higher Moments and Applications in Number Theory

In this talk, we investigate double sums involving monomials over intervals and arbitrary subsets of finite fields. We establish bounds that both generalise and, in certain ranges, sharpen previously known results. Furthermore, we study higher moments of double exponential sums with monomials over arbitrary sets and intervals in finite fields. Our approach relies on Fourier analysis on finite additive groups to obtain precise moment estimates. This area of research originates in the work of Heath-Brown, particularly in his study of least square-free numbers in arithmetic progressions, and has since developed into an important theme in analytic number theory over finite fields.