On the determinants of matrices with elements from arbitrary sets

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