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.

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In this paper

The sharp support threshold

Let $S \subseteq N_{> 0}$ and $c > 0$ . Define $E (S, c)$ to be the set of sums

\begin{matrix} (1) & x = \sum_{n \geq 1} \frac{e_{n}}{n!}, e_{n} \in Z_{\geq 0}, e_{n} = 0 (n \notin S), e_{n} \leq n^{c} eventually, \end{matrix}

subject to

\begin{matrix} (2) & for every q \geq 1, q ∣ e_{n} for all sufficiently large n . \end{matrix}

The cutoffs and the finite initial digits in this definition may depend on $x$ . All series converge, since finitely many unrestricted digits do not affect convergence. A permitted position need not carry a nonzero digit.

Theorem 3.2. If $S$ is finite, $E (S, c)$ is countable. If $S$ is infinite, enumerate it as $n_{0} < n_{1} < \dots$ .

For $0 < c \leq 1$ , $E (S, c)$ is null and meagre.
For $c > 1$ , $E (S, c)$ contains a nondegenerate interval if and only if
$\begin{matrix} (3) & n_{j} - n_{j - 1} < c for all sufficiently large j . \end{matrix}$
If this condition fails, $E (S, c)$ is null and meagre.

When (3) holds, one may require $e_{n} = 0$ before any prescribed cutoff and impose both

\begin{matrix} (4) & q ∣ e_{n}, q ∣ \sum_{k < n} e_{k} \end{matrix}

for all sufficiently large $n$ , with cutoffs depending on $q$ but independent of $x$ throughout the constructed interval.

The theorem is an ordinary mathematical proof. The associated formal sources check the general digit-feedback construction and, separately, the common-divisor carry obstruction of Section 3.6; they do not formalise this gap classification or its measure argument. The source and attribution account is in Section 12.

For example, with allowance $n^{2}$ , every second position suffices without congruences. Under (2), infinitely many omitted positions already force a null set. With allowance $n^{2.01}$ , every second position again suffices, even under (4). Thus the strict inequality in (3) is essential.

Corollary 3.3. For $c > 1$ , the least possible asymptotic density of a fixed permitted support that fills an interval under (2) is $1 / (⌈ c ⌉ - 1)$ . Every such support has lower density at least this value, and an arithmetic progression attains it. A density bound alone is not sufficient: even rare gaps of length $⌈ c ⌉$ prevent interval filling if they occur infinitely often.

Proof. Put $d = ⌈ c ⌉ - 1$ . Eventual gaps at most $d$ give $\underset{N \to \infty}{lim inf} | S \cap [1, N] | / N \geq 1 / d$ . An arithmetic progression of step $d$ satisfies the theorem. ◻

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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