Trace of Products in Finite Fields from a Combinatorial Point of View

The notion of digits in finite fields was introduced a few years ago as an attempt to deliver insight in related yet unresolved questions over the natural numbers. Several such intractable questions are related to the sum of digits function, which assigns to every natural number the sum of its digits. Its analogue over finite fields, which is a map from F_q to F_p, q = p h where p prime and h ≥ 2, has been studied by several authors. In particular, Cathy Swaenepoel [J. Number Theory, 189(2018), pp. 97-114] investigated the sum of digits of products of field elements. The main techniques involved were estimates on certain character sums and Gaussian sums over F_q and F_p. In this paper, we extend and generalize these results using a different approach, based on spectral graph theory, without any reference to character theory.