Riemann-profile transference for discrete entropy monotonicity

Definitions and mathematical statement

Let $G$ be a torsion-free abelian group: if $mg=0$ for an integer $m\geq1$, then $g=0$. Let $X$ have a discrete distribution of finite Shannon entropy on $G$, with countable support, and let $X_1,X_2,\ldots$ be independent copies. Write $S_j=X_1+\cdots+X_j$ and \[ H(Y)=-\sum_y\mathbb P(Y=y)\log_2\mathbb P(Y=y), \qquad 0\log_2 0=0. \] The entropies of all the finite sums are finite, since $H(S_j)\leq jH(X)$.

The proposed entropy-increment inequality is the following uniform asymptotic assertion. For every fixed integer $n\geq2$ and every $\varepsilon>0$, does there exist $h_0(n,\varepsilon)$ such that, for every such group and distribution with $H(X)\geq h_0(n,\varepsilon)$, \[ H(S_n)-H(S_{n-1}) \geq\frac12\log_2\!\left(\frac n{n-1}\right)-\varepsilon? \] The threshold must be independent of the particular distribution and ambient group. The asymptotic parameter is the entropy of $X$, while $n$ remains fixed; the question does not assert a uniform error as $n$ tends to infinity. The case $n=2$ is the entropy-doubling result motivating the workshop question, and the requested extension concerns subsequent increments. Torsion-freeness is essential to the formulation: uniform distributions on large finite subgroups would otherwise have arbitrarily large entropy but no entropy increase under addition.

Short English statement

Must each fixed later sum of independent copies gain at least the corresponding Gaussian entropy increment when the original discrete entropy is large?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000904 (AIM-COMBINATORICS-0029), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Additive combinatorics and its applications, item 2.4. Original workshop question underlying AIM-COMBINATORICS-0029. http://aimpl.org/addcombapp/2/.