Dimension-free discretizations of the uniform norm by small product sets

Let $f$ be an analytic polynomial of degree at most $K-1$. A classical inequality of Bernstein compares the supremum norm of $f$ over the unit circle to its supremum norm over the sampling set of the $K$-th roots of unity. Many extensions of this inequality exist, often understood under the umbrella of Marcinkiewicz-Zygmund-type inequalities for $L^p,1\le p\leq \infty$ norms. We study dimension-free extensions of these discretization inequalities in the high-dimension regime, where existing results construct sampling sets with cardinality growing with the total degree of the polynomial. In this work we show that dimension-free discretizations are possible with sampling sets whose cardinality is independent of $\deg(f)$ and is instead governed by the maximum individual degree of $f$; i.e., the largest degree of $f$ when viewed as a univariate polynomial in any coordinate. For example, we find that for $n$-variate analytic polynomials $f$ of degree at most $d$ and individual degree at most $K-1$, $\|f\|_{L^\infty(\mathbf{D}^n)}\leq C(X)^d\|f\|_{L^\infty(X^n)}$ for any fixed $X$ in the unit disc $\mathbf{D}$ with $|X|=K$. The dependence on $d$ in the constant is tight for such small sampling sets, which arise naturally for example when studying polynomials of bounded degree coming from functions on products of cyclic groups. As an application we obtain a proof of the cyclic group Bohnenblust-Hille inequality with an explicit constant $O(\log K)^{2d}$.