Problem 319

Erdős Problem #319: declared status 'open'. Formalized: yes. What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that$$\sum_{n\in A}\frac{\delta_n}{n}=0$$and$$\sum_{n\in A'}\frac{\delta_n}{n}\neq 0$$for all non-empty $A'\subsetneq A$? Current best: Adenwalla has observed that a lower bound of$$\lvert A\rvert\geq (1-\tfrac{1}{e}+o(1))N$$follows from the main result of Croot [Cr01], which states that there exists some set of integers $B\subset [(\frac{1}{e}-o(1))N,N]$ such that $\sum_{b\in B}\frac{1}{b}=1$. Since the sum of $\frac{1}{m}$ for $m\in [c_1N,c_2N]$ is asymptotic to $\log(c_2/c_1)$ we must have $\lvert B\rvert \geq (1-\tfrac{1}{e}+o(1))N$. Prize: no. Tags: number theory, unit fractions.