The number of multiplicative Sidon sets of integers

TL;DR

This paper determines the asymptotic number of multiplicative Sidon subsets of {1,...,n}, resolving a longstanding enumeration problem and extending results to generalized multiplicative Sidon sets.

Contribution

It provides the first precise asymptotic count of multiplicative Sidon sets and their generalizations, including explicit formulas and elementary proofs.

Findings

Number of multiplicative Sidon subsets is $T(n) imes 2^{ heta(n)}$ with specified $T(n)$.

The order of the lower order term in the exponent is precisely determined.

Extension to multiplicative $k$-Sidon sets with explicit constants.

Abstract

A set $S$ of natural numbers is multiplicative Sidon if the products of all pairs in $S$ are distinct. Erd\H{o}s in 1938 studied the maximum size of a multiplicative Sidon subset of $\{1,\ldots, n\}$, which was later determined up to the lower order term: $\pi(n)+\Theta(\frac{n^{3/4}}{(\log n)^{3/2}})$. We show that the number of multiplicative Sidon subsets of $\{1,\ldots, n\}$ is $T(n)\cdot 2^{\Theta(\frac{n^{3/4}}{(\log n)^{3/2}})}$ for a certain function $T(n)\approx 2^{1.815\pi(n)}$ which we specify. This is a rare example in which the order of magnitude of the lower order term in the exponent is determined. It resolves the enumeration problem for multiplicative Sidon sets initiated by Cameron and Erd\H{o}s in the 80s. We also investigate its extension for generalised multiplicative Sidon sets. Denote by $S_k$, $k\ge 2$, the number of multiplicative $k$-Sidon subsets of $\{1,\ldots, n\}$. We show that $S_k(n)=(\beta_k+o(1))^{\pi(n)}$ for some $\beta_k$ we define explicitly. Our proof is elementary.