Extremal densities for forbidden configurations in S-smooth numbers
Pith reviewed 2026-05-10 09:35 UTC · model grok-4.3
The pith
The largest subset of S-smooth numbers up to X avoiding any full set {n, p1 n, ..., pr n} has size exactly r/(r+1) times the total count of S-smooth numbers up to X, plus an error of size O((log X)^{r-1}).
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We prove that F_S(X) equals r/(r+1) times Ψ_S(X) plus an error O_S of (log X) to the power r-1 as X tends to infinity, where Ψ_S(X) counts the S-smooth integers up to X. Equivalently, the extremal function f_S(k) on the first k S-smooth numbers satisfies f_S(k) equals r/(r+1) times k plus an error O_S of k to the power (r-1)/r. The proof proceeds by reducing the smooth-number problem to the known theory of forbidden configurations on the full interval [1,N] and controlling the error that arises from the distribution of smooth numbers.
What carries the argument
The extremal function F_S(X) measuring the largest subset of S-smooth numbers up to X that contains no complete set {n, p1 n, ..., pr n}, together with its reduction to the classical extremal problem on all positive integers.
If this is right
- The density constant α_S admits an explicit representation in terms of the successive increments of the extremal function f_S.
- Nested computable upper and lower bounds become available for α_S.
- A recursive formula holds for the reciprocal tail sums taken over S-smooth numbers.
- For the special case S = {2,3} an explicit formula for the tail is obtained together with two structural properties of the optimal sets.
- Rational sums of reciprocals with S-smooth denominators need not arise from eventually periodic binary sequences.
Where Pith is reading between the lines
- The result indicates that the r/(r+1) density is controlled by local multiplicative relations rather than the global spacing of the ambient set.
- Similar density statements may hold for other thin multiplicative subsets such as prime powers or integers with restricted prime factors.
- The computable bounds supplied by the reduction allow numerical approximation of the constant α_S for concrete choices of S.
- The non-periodicity example shows that tail sums over smooth denominators can realize a wider class of values than those generated by periodic sequences.
Load-bearing premise
The reduction of the forbidden-configuration problem from S-smooth numbers to the classical setting on the full interval [1,N] is valid and the error terms coming from the distribution of smooth numbers do not change the leading r/(r+1) density term.
What would settle it
For a fixed S with r primes, compute or bound F_S(X) for a sequence of X going to infinity and check whether the absolute difference between F_S(X) and r/(r+1) times Ψ_S(X) stays below any constant times (log X) to the power r-1 for all sufficiently large X.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that for a finite set S of r distinct primes, the maximum size F_S(X) of a subset of S-smooth integers up to X avoiding the configuration {n, p_1 n, ..., p_r n} satisfies F_S(X) = (r/(r+1)) Ψ_S(X) + O_S((log X)^{r-1}) as X → ∞, with an equivalent statement for the extremal function f_S(k) on the first k S-smooth numbers. It relates the problem to the classical extremal configuration problem on [1,N], derives a representation of the density α_S via increments of f_S together with nested computable bounds and a recursive tail formula, shows that rational reciprocal sums over S-smooth denominators need not come from eventually periodic binary sequences, and for S={2,3} gives an explicit tail formula plus two structural propositions on optimal sets.
Significance. If the main asymptotic holds, the result supplies a precise extension of classical forbidden-configuration extremal theory to the multiplicative setting of S-smooth numbers, with an error term that matches the known discrepancy in the lattice-point count for Ψ_S(X). The explicit tail formula and structural propositions for S={2,3}, the recursive representation of the reciprocal tail, and the computable bounds on α_S are concrete strengths that make the density constant accessible for small r. The work cleanly separates the combinatorial density r/(r+1) (which follows from the structure of the exponent lattice) from the analytic error contributed by the distribution of smooth numbers.
major comments (2)
- [Section 3 (relation to the classical problem)] The reduction from the S-smooth problem to the classical extremal problem on [1,N] is invoked to obtain the representation of α_S and the auxiliary bounds, yet the manuscript does not explicitly verify that the O((log X)^{r-1}) discrepancy in Ψ_S(X) does not propagate into the leading coefficient when the forbidden configuration is imposed only on the smooth subset; a short paragraph comparing the two error regimes would strengthen the argument.
- [Section 5 (S={2,3} case)] For the structural propositions in the case S={2,3}, the explicit tail formula is stated but the proof that the optimal set is eventually periodic in the exponent lattice (or satisfies the claimed recursive relation) is only sketched; the induction step on the tail sum should be written out in full to confirm that the O(k^{(r-1)/r}) error is preserved.
minor comments (3)
- [Introduction] The notation α_S for the density constant is introduced after the main theorem; moving its definition to the introduction would improve readability.
- [Section 4] The statement that rational reciprocal sums over S-smooth denominators need not arise from eventually periodic binary sequences is interesting but lacks a concrete numerical example; adding one small explicit counter-example would make the claim immediately verifiable.
- [Section 2] A few citations to the classical literature on forbidden configurations (e.g., the original Erdős–Turán or Füredi–Sudakov results) appear only in the bibliography; brief parenthetical references in the text when the classical theory is applied would help readers.
Simulated Author's Rebuttal
We thank the referee for the careful reading, positive evaluation, and constructive suggestions. We address each major comment below and will make the indicated revisions to strengthen the manuscript.
read point-by-point responses
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Referee: [Section 3 (relation to the classical problem)] The reduction from the S-smooth problem to the classical extremal problem on [1,N] is invoked to obtain the representation of α_S and the auxiliary bounds, yet the manuscript does not explicitly verify that the O((log X)^{r-1}) discrepancy in Ψ_S(X) does not propagate into the leading coefficient when the forbidden configuration is imposed only on the smooth subset; a short paragraph comparing the two error regimes would strengthen the argument.
Authors: We agree that an explicit comparison would improve clarity. The leading coefficient r/(r+1) is determined purely by the combinatorial structure of the forbidden configuration within the exponent lattice and is inherited directly from the classical extremal density on [1,N] via the reduction; it is independent of the distribution of S-smooth numbers. The O((log X)^{r-1}) term is the standard discrepancy in the count Ψ_S(X) itself. Because the classical problem on [1,N] admits an exact leading density (with o(N) error) and the reduction maps subsets of S-smooth numbers to subsets of [1,N] while preserving the relative ordering and the forbidden-configuration condition up to a multiplicative factor absorbed into the smooth-number count, the discrepancy remains confined to the error term and does not affect the leading coefficient. We will insert a short paragraph in Section 3 that compares the two error regimes and confirms that the O((log X)^{r-1}) term is preserved under the reduction. revision: yes
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Referee: [Section 5 (S={2,3} case)] For the structural propositions in the case S={2,3}, the explicit tail formula is stated but the proof that the optimal set is eventually periodic in the exponent lattice (or satisfies the claimed recursive relation) is only sketched; the induction step on the tail sum should be written out in full to confirm that the O(k^{(r-1)/r}) error is preserved.
Authors: We thank the referee for this observation. The structural propositions for S={2,3} are established by induction on the tail sum in the exponent lattice, where the recursive relation follows from the optimality condition at each step. The manuscript provides a sketch of the base case and the inductive step, but we acknowledge that expanding the induction in full will make the argument self-contained and will explicitly verify that the O(k^{(r-1)/r}) error is preserved at every stage of the recursion. We will write out the complete induction argument in the revised Section 5. revision: yes
Circularity Check
No significant circularity; derivation applies external classical theory to independent smooth-number count
full rationale
The central asymptotic F_S(X) = r/(r+1) Ψ_S(X) + O((log X)^{r-1}) is obtained by transferring the known density r/(r+1) from the classical forbidden-configuration problem on [1,N] to the S-smooth integers via the independently established counting function Ψ_S(X). The coefficient itself follows from combinatorial arguments on the exponent lattice N_0^r (explicitly verified for r=1 and via tail formulas for r=2), not from any definition or fit internal to the paper. Auxiliary relations to the full-interval problem and recursive tail formulas are derived after the leading term and serve only for computable bounds and representations of α_S; they do not redefine or re-predict the density. No self-citations are load-bearing, no parameters are fitted then re-derived as predictions, and the error term matches the known lattice-point discrepancy of Ψ_S(X) without circular adjustment. The derivation is therefore self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The classical theory of forbidden configurations applies directly to subsets of S-smooth numbers after suitable embedding into [1,N].
Reference graph
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discussion (0)
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