On the determinants of matrices with elements from arbitrary sets

Abstract

Recently there have been several works estimating the number of $$ matrices with elements from some finite sets $$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial upper bound $$, where $$ is the cardinality of $$. Here we show that even for arbitrary sets $$, some recent results from additive combinatorics enable us to obtain a stronger bound with a power saving.