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arxiv: 2606.24972 · v1 · pith:I5VDY4DKnew · submitted 2026-06-23 · 🧮 math.NT

Positive dyadic density for rational weighted binary expansions

Han Wang , Jose Maria Grau Ribas This is my paper

Pith reviewed 2026-06-25 22:53 UTC · model grok-4.3

classification 🧮 math.NT
keywords binary expansionsrational weighted sumspositive dyadic densitycarry recurrenceErdős problem 260irrational seriessupport setsdyadic blocks
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The pith

If a weighted binary series sums to a rational P/Q, its support set has positive density in every large dyadic interval [X,2X].

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that whenever the series sum n d_n 2^{-n} equals a fixed rational P/Q with infinite support S, the counting function A_S obeys A_S(2X) minus A_S(X) at least c_Q X for all sufficiently large dyadic X, with c_Q depending only on Q. The argument relies solely on the carry recurrence that the rational equality forces on the partial sums. This immediately implies that any strictly increasing sequence a_n satisfying a_n over n tends to infinity produces an irrational value for the substituted series sum a_n 2^{-a_n}, which settles Erdős Problem 260. A reader cares because the result converts an arithmetic constraint (rationality) into a uniform lower bound on the distribution of 1-bits at dyadic scales.

Core claim

Rationality of the sum forces an integral carry recurrence on the partial sums. Sparse dyadic blocks then produce a positive lower bound on an integrated high-excess area, while a weighted stopping-time argument supplies the matching upper bound; the local carry geometry required for the upper bound is captured by the four estimates of complete-lap mass balance, total-support summation, fixed-pin confinement, and class-one realization. The net effect is that the support S cannot be thinner than linear density c_Q on large dyadic intervals.

What carries the argument

The integral carry recurrence forced by equality to the rational P/Q.

If this is right

  • Any sequence with a_n/n tending to infinity yields an irrational substituted series.
  • The lower density constant c_Q depends only on the denominator Q and is positive.
  • The support S of any such rational series must intersect every sufficiently long dyadic interval in a positive fraction of its length.
  • Finite-support series are the only ones that can be both rational and eventually zero on dyadic blocks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same carry-recurrence method might adapt to show positive density for sums equaling quadratic irrationals in other bases.
  • One could test whether the minimal c_Q is achieved by the series with smallest denominator.
  • The result suggests that rationality imposes a rigid periodic structure on the carry sequence that prevents arbitrarily large gaps at every scale.
  • Extending the four local estimates to non-dyadic intervals could yield an ordinary asymptotic density lower bound.

Load-bearing premise

That equaling a rational forces a carry recurrence whose local geometry is bounded by the four listed estimates.

What would settle it

An explicit rational P/Q together with a dyadic X large enough that the number of 1-bits of the series in (X,2X] falls below c_Q X.

read the original 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 Erd\H{o}s 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves that if sum_{n>=1} n d_n 2^{-n} = P/Q in Q with infinite support S = {n : d_n=1}, then S has positive lower dyadic density: there exists c_Q >0 (depending only on Q) such that A_S(2X) - A_S(X) >= c_Q X for all sufficiently large dyadic X, where A_S(X) counts elements of S up to X. This implies that any strictly increasing sequence a_n with a_n/n -> infinity yields an irrational sum sum a_n 2^{-a_n}, resolving Erdős Problem 260. The argument derives the density from the integral carry recurrence forced by rationality, using sparse dyadic blocks for a lower bound on integrated high-excess area and a weighted stopping-time argument for the upper bound; the local geometry is controlled by four estimates (complete-lap mass balance, total-support summation, fixed-pin confinement, class-one realization).

Significance. If the central claim holds, the result is significant: it settles an open problem of Erdős on irrationality criteria for lacunary series and gives a uniform positive-density statement for the support of any rational weighted binary sum. The proof strategy relies only on the forced integral carry recurrence without introducing free parameters or self-referential fits, and isolates the needed local bounds in four explicit estimates; these features strengthen the contribution if the derivations are complete.

major comments (2)
  1. [Abstract and §3 (carry recurrence section)] The abstract states that the lower bound follows from an integrated high-excess area over sparse blocks and the upper bound from a weighted stopping-time argument, but the manuscript must explicitly verify that the four listed estimates (complete-lap mass balance, total-support summation, fixed-pin confinement, class-one realization) are proved in full and that their constants depend only on Q (not on the particular rational or the block).
  2. [§4 (density extraction)] The claim that the density holds for every sufficiently large dyadic X requires a uniform lower bound c_Q independent of the starting point of the dyadic interval; the stopping-time argument must be checked to ensure it produces a positive density that does not deteriorate with the scale.
minor comments (2)
  1. [Introduction] Notation for dyadic X (powers of 2) should be defined explicitly at first use, and the dependence of c_Q on Q should be stated more quantitatively if possible.
  2. [§2] The manuscript would benefit from a short table or diagram illustrating one period of the carry recurrence for a small Q (e.g., 1/3) to make the four estimates more concrete.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The two major comments concern explicit verification of the four estimates in §3 and uniformity of the density bound in §4. We address each point below and will incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Abstract and §3 (carry recurrence section)] The abstract states that the lower bound follows from an integrated high-excess area over sparse blocks and the upper bound from a weighted stopping-time argument, but the manuscript must explicitly verify that the four listed estimates (complete-lap mass balance, total-support summation, fixed-pin confinement, class-one realization) are proved in full and that their constants depend only on Q (not on the particular rational or the block).

    Authors: Each of the four estimates is proved in full in §3 directly from the integral carry recurrence determined by the fixed denominator Q. The proofs rely on finite-state properties of the carry process and do not introduce parameters depending on the specific block or on any particular rational beyond Q. We will add a short concluding paragraph to §3 that lists the four estimates, cites their derivations, and states that all constants depend only on Q. revision: yes

  2. Referee: [§4 (density extraction)] The claim that the density holds for every sufficiently large dyadic X requires a uniform lower bound c_Q independent of the starting point of the dyadic interval; the stopping-time argument must be checked to ensure it produces a positive density that does not deteriorate with the scale.

    Authors: The weighted stopping-time argument in §4 yields a lower bound c_Q that is independent of the starting point and of the dyadic scale, because the weights are derived from the Q-dependent carry recurrence, which is invariant under dyadic rescaling. The sparse-block lower bound on integrated excess area is likewise scale-invariant. We will insert a brief remark in §4 confirming this uniformity and noting that the resulting density bound does not deteriorate with scale. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from carry recurrence

full rationale

The paper establishes positive dyadic density for the support S of a binary expansion summing to rational P/Q by invoking the integral carry recurrence forced by rationality, then bounding local geometry via the four stated estimates (complete-lap mass balance, total-support summation, fixed-pin confinement, class-one realization) to obtain integrated high-excess area lower bounds and weighted stopping-time upper bounds. No step reduces the target density by construction to a fitted parameter, self-definition, or self-citation chain; the argument is a direct analytic consequence of the recurrence and does not rename or smuggle prior results. This is the standard non-circular extraction of density from a bounded-state recurrence.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that rationality of the weighted sum imposes a specific integral carry recurrence whose geometry yields the stated density bounds.

axioms (1)
  • domain assumption Rationality of sum n d_n 2^{-n} forces an integral carry recurrence in the binary addition process.
    Explicitly identified in the abstract as the sole mechanism used in the proof.

pith-pipeline@v0.9.1-grok · 5757 in / 1208 out tokens · 25257 ms · 2026-06-25T22:53:58.521909+00:00 · methodology

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