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\bigl((\log X)^{r-1}\bigr) $ as $X \to \infty$, and equivalently that $ f_S(k)=\frac{r}{r+1}k+O_S\bigl(k^{(r-1)/r}\bigr) $ 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.