Extremal densities for forbidden configurations in $S$-smooth numbers

Abstract

Let $S=\{p_1,\dots ,p_r\}$ be a finite set of distinct primes, let ${\Psi }_S(X)$ be the number of $S$-smooth integers not exceeding $X$, and let $F_S(X)$ be the maximum size of a subset of $M(S)\cap [1,X]$ containing no set $\{n,p_{1}n,\dots ,p_{r}n\}$. We prove that $F_S(X)=\frac{r}{r+1}{\Psi }_S(X)+O_S((\log X{)}^{r-1})$ as $X\to \infty$, and equivalently that $f_S(k)=\frac{r}{r+1}k+O_S\left(k^{(r-1)/r}\right)$ for the corresponding extremal function on the first $k$ $S$-smooth numbers. We also relate this problem to the analogous extremal problem on the full interval $[1,N]$. Using the classical theory of such forbidden configurations, we obtain a representation of the corresponding density constant ${\alpha }_S$ in terms of the increments of $f_S$, along with nested computable bounds and a recursive formula for the reciprocal tail over $S$-smooth numbers. We further show that rational reciprocal sums over $S$-smooth denominators need not arise from eventually periodic binary sequences. In the classical case $S=\{2,3\}$, we derive an explicit tail formula and prove two structural propositions for optimal sets.

arXiv comment14 pages