Reciprocal Mersenne Subseries

Will Cook

Long paper

Reciprocal Mersenne Subseries

Erdős #257 By Will Cook 30 September 2026 Authorship, AI use and citation

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Summary. A weighted summability criterion proves irrationality of reciprocal Mersenne subseries: if h is the part of a supported on a finite nonempty prime set P, every infinite A with ∑ $h (a) / (a (b^{h (a)} - 1))$ finite has irrational ∑ $(b^{a} - 1)^{- 1}$ . The base-two condition gives the conclusion at every integer base for every infinite subset, and permits an example with divergent reciprocal sum, while excluding full support and all odd exponents. The proof averages over a finite part of the support and over dyadic observation lengths. Erdős's weaker reciprocal-summable theorem is proved in full, and a common window extends both arguments to unions of weighted and positive-cover supports. Finite-denominator and achievement-set calculations formulate tests for the targets $1 / 2$ and $1 / 21$ ; those tests do not decide either target or the universal question. Prime support at base 2 and squarefree support at power-of-two bases are attributed as existing work.

This paper covers a claim-bounded problem-specific reasoning surface for Erdős #257: selected checked premises, finite computations, closed routes, surviving obligations, and their exact evidence bands.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #257, whose universal and half-value questions remain open.

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In this paper

Geometry and rational-target membership

At the first differing digit, $w_{n} > R_{n}$ prevents all later digits from cancelling the difference. This proves uniqueness of the support coding. Uniform convergence on the product of the two-point digit spaces gives compactness. These facts have distinct uses below: uniqueness recovers a represented target by the greedy rule, while closedness turns a convergent sequence of finite subseries into a represented value.

Theorem 1.8 (Achievement-set geometry). $A$ is compact, closed, perfect, totally disconnected and nowhere dense, and $volume (A) = 1$ . Thus its measure is positive although it contains no interval. Its convex hull is $[0, E]$ , where $E = \sum_{n \geq 1} w_{n}$ . The positive-index digit coding onto $A$ is injective: each achievable real has exactly one support.

Theorem 1.9 (Support-restricted refinement). Use zero-based indices in this statement: coordinate $j \in N$ carries weight $w_{j + 1}$ . For $J \subseteq N$ , consider the sums that use only coordinates in $J$ . If $N ∖ J$ is finite, this achievement set has measure $2^{- | N ∖ J |}$ ; if infinitely many coordinates are omitted, its measure is zero. Injectivity survives every restriction; perfectness is proved when $J$ is infinite. No perfectness claim is made for finite $J$ , whose coding range is finite.

Hausdorff dimension of a restricted set.

For example, retaining the positive exponents $2, 4, 6, \dots$ gives a set of measure zero but Hausdorff dimension $1 / 2$ . More generally, let $J \subseteq {1, 2, \dots}$ and put

S_{J} = {\sum_{n \in J} ε_{n} w_{n} : ε_{n} \in {0, 1}}, s_{N} = # (J \cap {1, \dots, N}), d = \underset{N \to \infty}{lim inf} s_{N} / N .

Then $\dim_{H} S_{J} = d$ . Here is an ordinary proof, separate from the formal statements above. If $J$ is finite, both sides are zero, so suppose $J$ is infinite.

Write $w_{n} = 2^{- n} + δ_{n}$ , where $δ_{n} = \sum_{r \geq 2} 2^{- r n}$ . A geometric sum gives

\sum_{k > n} δ_{k} = \sum_{r \geq 2} \frac{2^{- r n}}{2^{r} - 1} \leq δ_{n} / 3 < δ_{n} .

The unique Mersenne digit coding therefore defines a map from $S_{J}$ onto the binary digit-restricted set $B_{J} = {\sum_{n \in J} ε_{n} 2^{- n}}$ by keeping the same digits. This map is $1$ -Lipschitz. In fact, at the first differing digit $n$ , the binary difference is nonnegative after orienting the larger digit first, and the Mersenne difference exceeds it by at least $δ_{n} - \sum_{k > n} δ_{k} > 0$ .

The $2^{s_{N}}$ Mersenne prefix intervals have length at most $R_{N} \leq 2^{1 - N}$ . For any $t > d$ , a subsequence with $s_{N} / N < (d + t) / 2$ makes their total $t$ -power length tend to zero. Hence $\dim_{H} S_{J} \leq d$ . For the reverse inequality, give the allowed binary digits independent probabilities $1 / 2$ . Each dyadic interval of length $2^{- N}$ has measure at most $2^{- s_{N}}$ ; endpoints have measure zero because $J$ is infinite. An interval $I$ with $2^{- N - 1} < | I | \leq 2^{- N}$ meets at most three such dyadic intervals. For $0 < s < d$ and all sufficiently large $N$ , its measure is therefore at most $3 \cdot 2^{- s_{N}} \leq 3 \cdot 2^{s} | I |^{s}$ . Summing this inequality over any sufficiently fine cover of $B_{J}$ gives a positive lower bound for the sum of its $s$ -power lengths. Thus $\dim_{H} B_{J} \geq s$ ; letting $s ↑ d$ proves $\dim_{H} B_{J} \geq d$ . A Lipschitz map cannot increase Hausdorff dimension, since it sends a cover to one of no larger diameters. The map above now yields $\dim_{H} S_{J} \geq d$ , as required. For $d = 0$ the lower bound is automatic.

Remark 1.10 (Achievement-set terminology and sources). We use the modern term achievement set for the subsum set that Kakeya’s 1914 note describes as a set of partial sums. For an infinite summable sequence of positive terms, Kakeya proves perfectness, and nowhere density under an all-index strict-tail condition; Kovač–Tao, Remark 4.1 verifies that strict-tail condition for the fixed-base Mersenne weights. Nitecki proves that a subsum set whose terms all exceed their tails is a Cantor set of measure $lim_{n} 2^{n} R_{n}$ , and credits this to Hornich [4]; this is the classical source for the topology and measure used here. The volume dichotomy, injectivity, and greedy-membership statements above and below are proved in Lean. The sparse power-series criteria of Kaneko–Suzuki–Tachiya (Theorems 1–3, pp. 3–5 of arXiv v1) assume sparse coefficient supports; their relation to the divisor-incidence coefficients of these subseries is recorded after Theorem 1.1.

Theorem 1.11 (Membership equals greedy survival; the fatal-gap dichotomy). For a real target $x \geq 0$ , let $r_{n} (x)$ be the remainder after the greedy rule has processed weights $w_{1}, \dots, w_{n}$ , and let $R_{n} = \sum_{j > n} w_{j}$ , with $r_{0} (x) = x$ and $R_{0} = E$ . Then

x \in A ⟺ x \geq 0 and r_{n} (x) \leq R_{n} for every n \geq 0.

For every $x \in [0, E]$ , nonmembership is equivalent to a finite strict gap between the two next-prefix intervals; see Observation 9.44.

Two boundary degeneracies are excluded at every rank, unconditionally: $r_{k - 1} (1 / 2) \neq w_{k}$ by the odd-denominator argument above, and $r_{k - 1} (1 / 2) \neq R_{k}$ because the tail differs from $E$ by a rational while $E$ is irrational. Both greedy decisions are therefore strict at every rank.

The following finite checks use exact integer arithmetic. Each has a published program and saved result, including the scan to row $200,000$ and the grouped computation to depth $13,548,057$ . Decimal margins summarise those integer data and give no asymptotic estimate. Reset margins are measured by $\log_{2} | rem (r + 1) - 2^{r + 1} | - (r + 5) / 2$ . The separate real-greedy scan uses $\log_{2} (4^{k} (r_{k - 1} (1 / 2) - w_{k}))$ at a take and $\log_{2} (4^{k} (R_{k} - r_{k - 1} (1 / 2)))$ at a safe skip. These are different normalisations, not one common margin statistic. Truncation rungs $J = 3, \dots, 22$ all proved to survive (not merely computed). Seam orbit certified to row $200,000$ : branch counts $R : M : U = 100197 : 49899 : 49898$ , the target inequality failing at exactly one reset row $6 \leq r \leq 200,000$ , namely $r = 7$ ; maximal pure-R run $19$ , following the reset at row $158,096$ .

Independently, rows $6 \leq n \leq 2500$ : $1209$ resets, zero classification anomalies, minimum in-scope margin $+ 1.1119$ bits at row $14$ growing to $\approx 1246$ bits at row $2499$ (mean $627$ ), never re-approaching the threshold.

Sharp tail margin, ranks $2$ to $3000$ : no fatal skip, take fraction $0.4992$ , tightest take-margin $+ 0.0340$ bits at rank $7$ (the near-tie $1 / 126 > 1 / 127$ ). In the stated units this is $\log_{2} (4^{7} (1 / 126 - 1 / 127)) = 0.034035 \dots$ . It is the smallest take-margin reported in this finite scan. The finite greedy prefix through $m = 200,000$ , certified rank by rank with integer interval bounds (saved run), satisfies $0 \leq 1 / 2 - \sum_{d \in G \cap [1, m]} w_{d} < 2^{- 199999}$ . The sum here is finite; this statement does not evaluate the infinite greedy sum.

A further exact finite computation, not used in the arguments here, regenerates the forced half-prefix support to depth $13,548,057$ (program, saved run). The reset rows $13$ – $30$ are also verified in Lean by direct evaluation (seamUpperResetDyadicBandEscape_through_thirty).

Remark 1.12 (Literature, exactly). $E$ is irrational [1]. Zudilin proves the irrationality measure bound $μ (E) \leq 2.46497868 \dots$ [9], correcting an earlier computation (see the remark at the end of his Section 3, p. 159). Transcendence of $E$ is unknown [7], and the Erdős–Graham conjecture that every such subset sum is irrational, i.e. (U), is itself open. For the binary digits of $E$ , Erdős’s proof produces arbitrarily long runs of zeros. Crandall asked whether the block 11 occurs infinitely often in the binary expansion of $E$ , and Campbell proves that it does [10].

The digit results concern particular patterns. An irrationality-measure bound controls lengths: a run of $L$ equal digits after position $N$ gives a dyadic approximation to $E$ with denominator at most $2^{N}$ , hence $L < (μ (E) - 1 + ε) N$ for all sufficiently large $N$ . The reset argument needs a run bound for its inhomogeneous recurrence, not for the binary digits of $E$ ; no implication from the cited digit results to that bound is proved here. Such a bound is an input to this particular approach to (H), not a necessary ingredient of every possible proof of membership.

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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