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.
From: Han Wang [view email]
[v1]
Tue, 23 Jun 2026 11:45:11 UTC (100 KB)
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