On the number of permutation-twisted dot products

Let $\mathbb{K}$ be a field of characteristic $0$. For each choice of distinct $a_1,\dots ,a_n\in \mathbb{K}$ and distinct $b_1,\dots ,b_n\in \mathbb{K}$, consider the sum $S={\sum }_{i=1}^n{a}_i{b}_{\pi (i)}$ as $\pi$ ranges over the permutations of $[n]$. We show that this sum always assumes at least $\mathrm{\Omega }(n^3)$ distinct values. This ``support’’ bound, which is optimal up to the value of the implicit constant, complements recent work of Do, Nguyen, Phan, Tran, and Vu, and of Hunter, Pohoata, and Zhu on the anticoncentration properties of $S$ when $a_1,\dots ,a_n,b_1,\dots ,b_n$ are real and $\pi$ is chosen uniformly at random.