On the determinants of matrices with elements from arbitrary sets

Recently there have been several works estimating the number of nxn$n\times n$ matrices with elements from some finite sets X${\mathcal {X}}$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial upper bound OXn2-1$O\left(X<^>{n<^>2-1}\right)$, where X$X$ is the cardinality of X${\mathcal {X}}$. Here we show that even for arbitrary sets X subset of R${\mathcal {X}}\subseteq {\mathbb {R}}$, some recent results from additive combinatorics enable us to obtain a stronger bound with a power saving.