Problem #297: Let $N\geq 1$. How many $A\subseteq \{1,\ldots,N\}$ are...

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Progress

It was not even known for a long time whether this is $2^{cN}$ for some $c<1$ or $2^{(1+o(1))N}$. In fact the former is true, and the correct value of $c$ is now known.

• Steinerberger [St24] proved the relevant count is at most $2^{0.93N}$;


• Independently, Liu and Sawhney [LiSa24] gave both upper and lower bounds, proving that the count is\[2^{(0.91\cdots+o(1))N},\]where $0.91\cdots$ is an explicit number defined as the solution to a certain integral equation;


• Again independently this same asymptotic was proved (with a different proof) by Conlon, Fox, He, Mubayi, Pham, Suk, and Verstraëte [CFHMPSV24], who prove more generally, for any $x\in \mathbb{Q}_{>0}$, a similar expression for the number of $A\subseteq \{1,\ldots,N\}$ such that $\sum_{n\in A}\frac{1}{n}=x$;


• The above papers all appeared within weeks of each other in 2024; in 2017 a similar question (with $\leq 1$ rather than $=1$) was asked on MathOverflow by Mikhail Tikhomirov and proofs of the correct asymptotic were sketched by Lucia, RaphaelB4, and js21.

See also [362].