A question of Erdős and Graham on Egyptian fractions

TL;DR

The paper resolves Erdős–Graham's question by showing that the number of representations of a fixed positive rational $x$ as a sum of distinct unit fractions with denominators up to $n$ grows as $2^{(c_x+o(1))n}$, with $c_x$ given by an entropy-driven integral and increasing in $x$. The authors develop a two-pronged approach: (i) an entropy-based counting framework to estimate the number of subsets with sum $\le x$, yielding the exponent $c_x$; (ii) an absorption technique that uses a reservoir and modular-inverse control (via sums modulo large primes and generalized arithmetic progressions) to upgrade the count to representations summing exactly to $x$. The resulting exponent $c_x=\int_0^1 h(1/(1+e^{\lambda/y}))\,dy$, where $\lambda$ solves $\int_0^1 1/(y(1+e^{\lambda/y}))\,dy = x$, implies $c_x\uparrow 1$ as $x\to\infty$ with $c_1\approx 0.91117$, thereby giving sharp asymptotics for all fixed $x$. The work blends entropy methods with a modular-absorption framework and introduces a key lemma on subset sums of modular inverses to enable the absorption step.

Abstract

Answering a question of Erdős and Graham, we show that for each fixed positive rational number $x$ the number of ways to write $x$ as a sum of reciprocals of distinct positive integers each at most $n$ is $2^{(c_x + o(1))n}$ for an explicit constant $c_x$ increasing with $x$.

Theorems & Definitions (18)

• Theorem 1
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof : Proof of Lemma \ref{['lem:main-ent']}
• Theorem 2
• Theorem 3
• Lemma 4
• proof