Reading Eight Erdős Problems Together

Will Cook

Cross-problem paper

Reading Eight Erdős Problems Together

Eight Erdős problems read together By Will Cook 22 September 2026 Authorship, AI use and citation

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Précis. One account of the mathematics developed across the programmes: an exact tail-capacity criterion under eventual congruences, its sharp factorial support-gap threshold, rational and irrational Lambert subsums across bases, and the limits of finite greedy tests and other irrationality methods. Full arguments, exact counterexamples, unsuccessful routes and attribution are retained together. The principal proofs are ordinary mathematics; cited Lean ingredients have their own stated scope. Historical novelty and independent expert review are not established.

This paper owns the synthesis exposition and ordinary proofs across the covered problems: capacity and congruence constructions, Lambert subsums, rational-point counts, method limits and their research record.

It is not authority for a solution to any original Erdős target, historical novelty, independent expert review, or a full Lean proof of the analytic capacity criterion or Lambert-chain theorem.

Start here, by the editorial route

In this paper

Lambert subsums across bases

For real $t > 1$ and a set $S$ of positive integers put

X_{S} (t) = \sum_{n \in S} \frac{1}{t^{n} - 1}, w_{n} = \frac{1}{t^{n} - 1}, R_{N} = \sum_{n > N} w_{n} .

Problem #257 asks whether $X_{S} (t)$ is irrational for every infinite $S$ at every integer $t \geq 2$ [6]; Problem #1049 asks about $X_{N_{> 0}} (t)$ at rational $t$ .

The comparison behind everything is stated for weights with multiplicities. Let $u_{n} > 0$ and integers $D_{n} \geq 1$ satisfy $\sum_{n} D_{n} u_{n} < \infty$ , and put

V = {\sum_{n \geq 1} ε_{n} u_{n} : ε_{n} \in {0, 1, \dots, D_{n}}}, C_{N} = \sum_{n > N} D_{n} u_{n} .

Theorem 6.1 (choices against contraction). (i) For every $N$ , the Lebesgue measure of $V$ is at most $C_{N} \prod_{n \leq N} (D_{n} + 1)$ . If $\underset{N}{lim inf} C_{N} \prod_{n \leq N} (D_{n} + 1) = 0$ then $V$ is null.

(ii) If $u_{n} \leq C_{n}$ for every $n > N_{0}$ , then for each choice of $ε_{1}, \dots, ε_{N_{0}}$ the set $V$ contains the interval $[μ, μ + C_{N_{0}}]$ , where $μ = \sum_{n \leq N_{0}} ε_{n} u_{n}$ .

(iii) For $u_{n} = β^{- n}$ with real $β > 1$ and $D_{n} = D$ , the bound in (i) tends to $0$ exactly when $D + 1 < β$ , and the hypothesis of (ii) holds exactly when $D + 1 \geq β$ .

Part (ii) is Kakeya’s covering argument and part (i) is the standard covering bound; both are classical [9][10][11], and Kovač and Tao give a scalar reciprocal-choice covering lemma [13]; their higher-dimensional approximation lemma is Lemma 7.2 of the same paper. We claim no novelty for Theorem 6.1. Its use here is to say which side each problem lies on.

Theorem 6.2 (the subseries across bases). Let $t > 1$ be real.

(a) If $t < 2$ , let $N_{0} \geq 0$ be least with $t^{- n} \leq 2 - t$ for all $n > N_{0}$ . Then the set of values $X_{S} (t)$ is the union of the intervals $[X_{F} (t), X_{F} (t) + R_{N_{0}}]$ over $F \subseteq {1, \dots, N_{0}}$ , where $X_{F} (t) = \sum_{n \in F} w_{n}$ ; in particular it contains $[0, R_{N_{0}}]$ . If moreover $t = a / b$ in lowest terms, then every rational in $[0, R_{N_{0}}]$ whose reduced denominator shares a prime factor with $a b$ equals $X_{S} (t)$ for some $S$ , and every such $S$ is infinite. There is an infinite $S \subseteq {2, 3, \dots}$ with $X_{S} (3 / 2) = 1 / 2$ .

(b) If $t = 2$ , every value has exactly one $S$ , and the set of values is a Cantor set of Lebesgue measure $1$ inside $[0, E]$ , where $E = \sum_{n \geq 1} (2^{n} - 1)^{- 1} = 1.6066951524 \dots$ .

(c) If $t > 2$ , the set of values is null.

Part (b) is proved in the companion note on #257, Section 7, from Hornich’s theorem as proved by Nitecki [33][9][10]; Kovač and Tao record the strict inequality $w_{N} > R_{N}$ and the Cantor-set conclusion for every fixed base $t \geq 2$ [13]. Parts (a) and (c) follow from Theorem 6.1 in a few lines. We have not found part (a) stated for non-integer bases and it may be known. It shows that the statement asked in #257 is false at every rational base below $2$ , so any proof at base $2$ must use more than the shape of the series.

Theorem 6.3 (divisibility chains). Let $S = {n_{1} < n_{2} < \dots}$ be infinite with $n_{j} ∣ n_{j + 1}$ for every $j$ .

(a) If $t = a / b > 1$ is rational in lowest terms and $a^{2} > b^{3}$ , then $X_{S} (t)$ is irrational.

(b) If the integer ratios $n_{j + 1} / n_{j}$ are eventually periodic, then $X_{S} (t)$ is transcendental for every algebraic real $t > 1$ .

For example, the chain $1, 2, 6, 12, 36, 72, \dots$ has alternating ratios $2, 3$ . Part (b) proves

\sum_{k \geq 0} (\frac{1}{(4 / 3)^{6^{k}} - 1} + \frac{1}{(4 / 3)^{2 \cdot 6^{k}} - 1}) is transcendental,

although $4^{2} < 3^{3}$ . Repeated blocks of ratios, rather than a stronger tail estimate, supply the functional equation used in this case. Part (b) is a direct corollary of the classical Mahler value theorem cited below; no historical novelty is claimed for this specialisation.

The hypothesis $a^{2} > b^{3}$ says $\log b / \log a < 2 / 3$ . It holds for every integer base, for $3 / 2$ , $5 / 2$ and $7 / 3$ , and fails for $4 / 3$ and $5 / 4$ . At integer bases part (a) is contained in the theorem of Erdős on supports with $\sum_{n \in S} 1 / n < \infty$ [8], proved in full in [33]. At base $3 / 2$ it sits inside the regime of Theorem 6.2(a): rational values occur there, and exact divisibility still forces irrationality. For comparison, the companion note on #1049 proves the irrationality of the full sum $X_{N_{> 0}} (a / b)$ when $\log b / \log a < 0.4056830213840605 \dots$ , using Zudilin’s linear forms [35], [14], and proves that the sufficient cutoff supplied by one integer-polynomial family with common leading degree, coefficient-height and decay bounds at every fixed real base $x > 1$ is at most $1 / 2$ [35]. This restriction does not exclude stronger estimates at a particular base or a different choice of family there. Thin supports reach further than the full sum because the denominators divide one another.

At base $2$ the greedy rule characterises membership and supplies finite certificates of nonmembership: starting from $r_{0} = x$ , take index $n$ when $r_{n - 1} \geq w_{n}$ and subtract. Since $w_{n} > R_{n}$ , a real $x \in [0, E]$ is a subsum if and only if no remainder falls strictly between $R_{n}$ and $w_{n}$ ; we say $x$ is rejected at step $n$ when that happens first at index $n$ .

Theorem 6.4 (the fixed-depth rational count). Fix $N \geq 1$ . Among the reduced fractions $p / q \in (0, E]$ with $q \leq Q$ , the proportion not rejected in the first $N$ steps tends to $2^{N} R_{N} / E$ as $Q \to \infty$ , with error $O (2^{N} \log Q / Q)$ . Consequently the upper limit of the proportion that are subsums is at most $1 / E = 0.62239 \dots$ .

The limit $2^{N} R_{N} / E$ does not depend on whether #257 is true. Agreement with this fixed-depth limiting proportion therefore does not establish membership. An exact computation in [36] first read a surviving share near $62 %$ as evidence that most such fractions are subsums; Theorem 6.4 is the correction. Section 6.4 states what remains.

Evidence.

Choices against contraction

The subseries across bases

Divisibility cuts at every algebraic base

Base two

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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