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

Prescribing a value and its derivatives

An integer factorial expansion can represent a real number while its digits eventually vanish modulo every fixed integer. Can the same digits prescribe several real quantities independently? Consider their exponential generating function. Its derivatives at $1$ are different weighted sums of the same digits, so this is simultaneous prescription, not several independent expansions.

For $c > 0$ , let $H_{c}$ consist of the entire functions

\begin{matrix} f (z) = \sum_{n \geq 1} e_{n} \frac{z^{n}}{n!}, e_{n} \in Z_{\geq 0}, \\ e_{n} \leq n^{c} eventually, (\forall q \geq 1) q ∣ e_{n} eventually . \end{matrix}

Polynomial coefficient growth implies locally uniform convergence of every derivative. Put $J_{d} (f) = (f (1), f^{'} (1), \dots, f^{(d - 1)} (1))$ and $J_{d, c} = {J_{d} (f) : f \in H_{c}} \subseteq R^{d}$ . These are Hurwitz functions: all derivatives at $0$ are integers. Our restrictions concern those integers; the prescribed derivatives are at $1$ .

Theorem 2.1 (Dimension and the sharp interpolation threshold). For every $c > 0$ and integer $d \geq 1$ ,

\dim_{H} J_{d, c} = min (c, d) .

If $c > d$ , this set contains a nonempty open subset of $R^{d}$ . If $c \leq d$ , it has $d$ -dimensional Lebesgue measure zero and is meagre; these conclusions already follow from eventual evenness of the coefficients. The dimension lower bound and the open-set conclusion can both be realised with $e_{n} \leq n^{c}$ at every position, an arbitrarily long initial zero segment, and congruence cutoffs common to the constructed family. The open set can be realised using only functions with eventually positive coefficients.

Theorem 2.2 (Rational derivatives force polynomiality). Let $d \geq 1$ , and let

f (z) = \sum_{n \geq 0} a_{n} \frac{z^{n}}{n!}, a_{n} \in Z, | a_{n} | = O (n^{d}) .

Assume that, for each $q \geq 1$ , $q ∣ a_{n}$ eventually. If $f (1), f^{'} (1), \dots, f^{(d - 1)} (1)$ are all rational, then $f$ is a polynomial. Consequently, a nonpolynomial member of $H_{c}$ with all these derivatives rational exists if and only if $c > d$ .

Thus if a nonpolynomial $f \in H_{2}$ has rational $f (1)$ , its derivative $f^{'} (1)$ is irrational. Increasing the allowance to $n^{2 + ε}$ permits both to be rational, for every $ε > 0$ . At $c = 2$ , the attainable pairs nevertheless form a full-dimensional null set. We do not assert that every vector is attained: nonnegative coefficients impose inequalities such as $f^{'} (1) \geq f (1)$ .

The identity behind the construction is

\begin{matrix} (J) & (z - 1) \sum_{n \geq 0} b_{n} \frac{z^{n}}{n!} = - b_{0} + \sum_{n \geq 1} (n b_{n - 1} - b_{n}) \frac{z^{n}}{n!} . \end{matrix}

Adding the left-hand side to a function leaves its value at $1$ unchanged and changes its derivative there by $\sum b_{n} / n!$ . Multiplication by $(z - 1)^{k}$ leaves the first $k$ entries of the derivative vector unchanged, at a cost of $k$ powers of $n$ in the coefficient allowance. The scalar construction of Theorem 3.1 supplies the free quantity. For necessity, eventual evenness reduces the number of prefixes by an exponential factor, which forces measure zero even at $c = d$ . Section 4 proves the theorem, including its dimension statement at the critical exponent. In the reverse direction, division by $(z - 1)^{d}$ converts vanishing derivatives into bounded integer coefficients. Eventual divisibility then forces them to vanish; this proves Theorem 2.2 in Section 5.

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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