Counting Admissible Sets of Integers: A Proof of the Erdős--Nicolas--Sárközy Conjecture

Source abstract

A finite set of positive integers is admissible if any two of its subsets with the same sum have the same cardinality. In 1991, Erdős, Nicolas and Sárközy conjectured that $\log a(N)\asymp\sqrt N$, where $a(N)$ denotes the number of nonempty admissible subsets of $\{1,\ldots,N\}$. We prove this conjecture by showing that there are absolute constants $c,C>0$ such that $\exp(c\sqrt N)\le a(N)\le\exp(C\sqrt N)$ for every integer $N\ge4$. The same bounds hold for the number of admissible sets with maximum $N$. For the upper bound, we show that every sufficiently large admissible set $A\subseteq\{1,\ldots,N\}$ contains a subset of proportional size whose doubling constant is $O(N/|A|^2)$.