Extremal densities for forbidden configurations in $S$-smooth numbers
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910138207567872 |
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| author | Veselinov, Nikola |
| author_facet | Veselinov, Nikola |
| contents | Let $S = \{p_1,\dots,p_r\}$ be a finite set of distinct primes, let $Ψ_S(X)$ 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}Ψ_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 $α_S$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15515 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extremal densities for forbidden configurations in $S$-smooth numbers Veselinov, Nikola Number Theory Combinatorics 11B75 (Primary) 11N25, 05D05 (Secondary) Let $S = \{p_1,\dots,p_r\}$ be a finite set of distinct primes, let $Ψ_S(X)$ 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}Ψ_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 $α_S$ 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. |
| title | Extremal densities for forbidden configurations in $S$-smooth numbers |
| topic | Number Theory Combinatorics 11B75 (Primary) 11N25, 05D05 (Secondary) |
| url | https://arxiv.org/abs/2604.15515 |