Rényi entropy conjecture for weighted Bernoulli sums

Prove that for every weight vector w in R^n, if X is uniformly distributed on {0,1}^n and S_w = <w,X>, then the zeroth-order Rényi entropy satisfies H_0(S_w) <= 2 H_infinity(S_w).

Background

The paper studies multiplicative comparisons between Rényi entropies of different orders for weighted sums of independent Bernoulli random variables. For a random vector X uniformly distributed on {0,1}n and a weight vector w, the weighted sum is S_w = <w,X>. The inequality H_0(S_w) <= 2 H_infinity(S_w) compares the logarithm of the number of attainable values of S_w with the min-entropy of its distribution, and is an anti-concentration statement.

The conjecture is presented as a strengthening of an earlier square-root-type estimate, H_0(S_w) <= C sqrt(n H_infinity(S_w)), and the paper proves logarithmic upper bounds instead. The authors also note that the constant 2 would be sharp if the conjecture holds, but the conjectured inequality itself is not established in the paper.

References

The following conjecture was communicated to us by Mokshay Madiman; an equivalent formulation appears in (see equation (1.2) therein). Let $X$ be a random vector uniformly distributed on ${0, 1}n$. For a weight vector $w\in Rn$, we define $S_w=\langle w, X\rangle$. For all $w\in\mathbb{R}n$, it holds that

H_0(S_w)\leq 2 H_\infty(S_w).

— Multiplicative comparisons of Rényi entropies for weighted Bernoulli sums  (2609.01529 - Li, 1 Sep 2026) in Conjecture 1.1, Section 1, Introduction