Wang 2026 positive dyadic density rational weighted binary

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Han Wang, Jose Maria Grau Ribas, Positive dyadic density for rational weighted binary expansions. arXiv preprint (2026). arXiv:2606.24972v1. The copy read for this card is v1 (2026-06-23); the arXiv record lists later versions v2 and v3 (2026-07-15, 2026-07-17) by Wang alone and v4 (2026-08-24), retitled Sparse Polynomial-Weighted Expansions. The labels below are those of v1.

Theorem 2.1 claims that if sum n d_n 2^{-n} = P/Q, in lowest terms with Q >= 1, with digits d_n in {0,1} and infinite support S, then A_S(2X) - A_S(X) >= c_Q X for all sufficiently large dyadic X, with c_Q depending only on Q. Corollary 2.2 deduces that every increasing sequence a_1 < a_2 < ... with a_n/n -> infinity gives an irrational series sum a_n 2^{-a_n}, which is Erdos Problem 260, since a_n/n -> infinity forces A_S(X) = o(X) and contradicts the density lower bound. The method is a contradiction on a single dyadic block whose only arithmetic input is the integer carry recurrence that a rational value imposes: a sparse block gives a pressure lower bound on an integrated area of high excess (Section 5), set against a weighted stopping-time upper bound, Theorem 6.4, whose local carry input reduces to four estimates (complete-lap mass balance, total-support summation, fixed-pin confinement, class-one realization) in Section 7 and Appendix C. Bearing on problem 260: the paper is the claimed full resolution. A concern has been noted about the step from Lemma B.6 (dyadic excess-bin domination) to Theorem 6.4, but its source (author, venue, date) is not recorded. The present reading confirmed only that those statements exist and are cited as inputs to the upper bound; the correctness of that dependency was not verified here.

Source: https://arxiv.org/abs/2606.24972. The arXiv record (https://arxiv.org/abs/2606.24972, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #260

Results to transcribe.

• Theorem 2.1: If sum n d_n 2^{-n} is rational with infinite support S, then A_S(2X) - A_S(X) >= c_Q X for all large dyadic X, with c_Q depending only on the denominator Q.
• Corollary 2.2: For positive integers a_1 < a_2 < ... with a_n/n -> infinity, the series sum a_n 2^{-a_n} is irrational, the statement of Erdos Problem 260.
• Theorem 6.4: Stopping-time upper bound: assuming the density deficit A_S(2X) - A_S(X) <= c_* X, for every xi > 0, after the constant hierarchy is chosen and then c_* is taken small enough, A_{r,0}(eps L) <= C_* xi r X |I_0| + C_Q c_* r X |I_0| + o(r X |I_0|).
• Lemma B.6: Dyadic excess-bin domination: with Y_nu = 2^nu Y_0 and the bins B_{k,nu} of thresholds T where the excess lies in [Y_nu, 2Y_nu), the area A_{s,j}(Y_0) is at most 2 sum_k sum_{nu >= 0} Y_nu |B_{k,nu}| + o(s X |I_j|), and the same sum over nu >= 1 is at most 2 A_{s,j}(Y_0) + o(s X |I_j|); cited as an input to the upper-bound ledger, and the step about which the unsourced concern above was noted.