Sparse Polynomial-Weighted Expansions

Han Wang

number theory

Sparse Polynomial-Weighted Expansions

arXiv:2606.24972

summary

The paper proves that if the weighted binary expansion defined by an infinite set of natural numbers S is rational, then S must occupy a positive proportion of every sufficiently large dyadic block, providing a local density obstruction that resolves Erdős Problem 260 as a corollary.

Abstract

Let $b \geq 2$ be an integer, let $p \in Q [x]$ be nonzero, and let $S \subseteq N$ be infinite. We prove that if $\sum_{n \in S} p (n) b^{- n}$ is rational, then, for every fixed $0 < θ \leq 1$ , there is a constant $c > 0$ such that $∣ S \cap (N, N + W] ∣ \geq c W$ for all sufficiently large $N$ and every $N^{θ} \leq W \leq N$ . Thus rationality forces positive lower density; if $S = {a_{1} < a_{2} < \dots}$ and $d = ° p$ , it also forces $a_{j + 1} - a_{j} \leq d lo g_{b} a_{j} + O (1)$ . In the binary linear case, the result implies the irrationality conjectured by Erdős whenever $a_{j} / j \to \infty$ . The proof turns rationality into an integer carry orbit of polynomial height. Repeated gap words lock the orbit onto rational polynomial graphs, while the normalized highest Newton coefficient evolves by an expanding affine map. Denominator preservation, polynomial sampling, and an interior--exterior count then rule out sparse polynomial-scale windows.

The formalization proof for lean4 is available at https://github.com/Hanziwww/erdos260

Topics & keywords

#weighted binary expansions #dyadic density #rationality #Erdős problem 260 #combinatorial number theoryη_Sdyadic blockreduced denominatorcarry analysisdensity obstruction