A Stretched-Exponential Bound for an Erdos--Graham Unit-Fraction Problem

Abstract

For a finite multiset $A$ of positive integers, write $\mathcal{R}(A)=\sum_{a\in A}a^{-1}$ and let $\varepsilon(A)$ be the distance from $1$ to the largest reciprocal subsum of $A$ that does not exceed $1$. Erdős and Graham proved that $\varepsilon(A)\ll K^{-2}$ whenever $\mathcal{R}(A)>K$, and asked whether one always has $\varepsilon(A)\leq \exp(-cK)$ for an absolute constant $c>0$. We prove the stretched-exponential estimate $$ \varepsilon(A)\leq \exp\bigl(-c\sqrt{K\log K}\bigr) $$ for all sufficiently large $K$.

Disclosure

“ρ(u) xu min u, β(P ) ≪ e2λ = β(P )2 , u P as required. AI Acknowledgement The author was assisted by GPT-5.5 Pro extensively during the development and writing of this manuscript. The main conceptual reductions and proof strategy, including the compression framework, the sparse activation method, and the box-moment estimates were developed by the”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-24T00:34:05.733Z

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