Positive dyadic density for rational weighted binary expansions

Abstract

Let \(P/Q\in\mathbb Q\), \(Q\ge1\), and suppose \[ \sum_{n\ge1} n d_n2^{-n}=P/Q,\qquad d_n\in\{0,1\}, \] has infinite support \(S=\{n:d_n=1\}\). We prove that \(S\) has positive density on all sufficiently large dyadic blocks: there is \(c_Q>0\), depending only on \(Q\), such that \[ A_S(2X)-A_S(X)\ge c_QX \] for every sufficiently large dyadic \(X\), where \(A_S(X)=\#(S\cap[1,X])\). Hence every increasing sequence \(a_1<a_2<\cdots\) with \(a_n/n\to\infty\) gives an irrational series \(\sum_{n\ge1}a_n2^{-a_n}\), settling Erdős Problem~260. The proof uses only the integral carry recurrence forced by rationality. Sparse dyadic blocks give a positive lower bound for an integrated high-excess area, while a weighted stopping-time estimate gives the matching upper bound. The local carry geometry needed for that upper bound is isolated in four estimates: complete-lap mass balance, total-support summation, fixed-pin confinement, and class-one realization.