How concentrated can a polynomial be under a random input? If X is a random vector and p(X) is a polynomial of degree d, can p(X) be very close to zero with high probability? Anti-concentration bounds say no: polynomials resist being concentrated. Glazer and Mikulincer (arXiv: 2603.22664) establish dimension-free anti-concentration results for polynomials on L^p balls with symmetric measures.
“Dimension-free” is the key qualifier. Most anti-concentration bounds degrade as the dimension increases — the polynomial can be more concentrated in higher-dimensional spaces. Dimension-free bounds say the concentration depends on the degree of the polynomial but not on the number of variables. This is the strongest type of anti-concentration result.
The method starts from variance: a lower bound on the variance of p(X) implies that p(X) can't be concentrated near any single value. For log-concave random vectors (the natural generalization of Gaussians), the authors show that variance bounds yield both small-ball estimates (p(X) is unlikely to be near zero) and Fourier decay estimates (the characteristic function of p(X) decays at a controlled rate).
The surprise: for uniform measures on L^p balls, obstructions to dimension-free bounds occur only when p = d is an even integer and the polynomial's coefficients resemble those of the normalized L^p norm itself. In all other cases, the bounds are dimension-free. The obstruction is a resonance between the polynomial degree and the geometry of the ball — when they align perfectly, concentration becomes possible.
The through-claim: polynomials spread because they're polynomials, not because of the space they live in. The dimension of the ambient space is irrelevant to concentration — what matters is the degree and the geometry of the underlying measure. The only exceptions are resonances: cases where the polynomial's structure perfectly matches the measure's geometry, allowing constructive concentration. Outside these resonances, polynomials are inherently anti-concentrated.
Glazer & Mikulincer, 2603.22664. Probability / anti-concentration / polynomials / log-concave measures / Carbery-Wright inequality.