Kaneko, Suzuki and Tachiya (2026): Sparse-Series Criteria

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corollary_2: Sufficiently sparse nonnegative algebraic-integer coefficients at a Pisot or Salem base exclude every algebraic degree up to a prescribed integer; satisfying the criterion for all degrees gives transcendence.

corollary_3: Nonnegative integer weights, infinitely many nonzero, whose partial sums up to x are at most a constant times x times a fixed power of log x give irrational series with totient or divisor-sum exponents at every integer base.

theorem_1: Sparse algebraic-integer coefficients at a Pisot or Salem base give a value outside the base field when their averaged complete tails are small and a positive sequence enters sufficiently large signed gaps.

theorem_2: A conjugate-size root bound and two auxiliary growth estimates imply the averaged complete-tail condition in the Pisot and Salem criterion.

theorem_3: Integer coefficients with root growth below the base, sparse support, sufficiently small mass and a support-interlacing condition give an irrational value at every integer base satisfying those hypotheses.

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Hajime Kaneko, Yuta Suzuki and Yohei Tachiya, Refinements of Erdős's irrationality criterion for certain sparse infinite series, arXiv:2601.20743v1, submitted 28 January 2026. The manuscript displays 29 January 2026. The paper was published online in International Journal of Number Theory on 26 June 2026 (doi:10.1142/S1793042126501137; Crossref record read). This record reads arXiv version one, whose labels and page numbers the result pages cite; the journal version was not compared.

The paper generalizes Erdos's 1957 irrationality criterion (his Lemma 4', quoted here as Theorem A) from integer bases to arbitrary Pisot or Salem numbers q of degree d. Theorem 1 is the fundamental criterion: for sequences a, b of algebraic integers in Q(q) with a(n) >= 0, growth, counting and tail-decay conditions (i)-(v) on auxiliary sequences x_n, y_n, z_n force sum (a(n)+b(n))/q^n to lie outside Q(q); Theorem 2 replaces the awkward tail condition (iv) by the more usable (iv-1)-(iv-3). Corollary 1 and Example 1 apply this to sumsets A+B of sparse index sets, and the abstract's headline application shows that for all integers t >= 2 and k >= 0 both sum d(n)^k/t^{sigma(n)} and sum d(n)^k/t^{phi(n)} are irrational, where d, sigma, phi are the divisor, divisor-sum and totient functions. The method is a quantitative Pisot/Salem conjugate-norm argument on truncated tails, and it supplies the proof Erdos omitted. For problem 68 the paper is real but off-point: it treats sparse arithmetic-function series, not the irrationality of sum 1/(n!-1). For problem 247 it is likewise adjacent, documenting the active criterion literature without addressing transcendence of sum 1/2^{a_n}.

Source: https://arxiv.org/abs/2601.20743.

Source identity

The copy read for this card is the 20-page version-one manuscript. The version and date were checked against the arXiv record on 17 September 2026. Printed and PDF page numbers agree. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2601.20743), every other right reserved.

Canonical results and proof routes

• Theorem 1, p. 3: an averaged complete-tail criterion at Pisot and Salem bases, with a sparsity hypothesis and an interlacing condition for signed support. Its proof on p. 11 uses the norm and tail lemmas on pp. 6–11.
• Theorem 2, p. 3: explicit coefficient and auxiliary growth conditions replace the averaged tail hypothesis. Lemma 4 and the proof are on pp. 11–13.
• Theorem 3, p. 5: the integer-base specialization removes the auxiliary factors involving the degree minus one. Both positive and signed supports must satisfy the stated sparsity bound.
• Corollary 2, p. 4: sufficiently sparse nonnegative coefficients exclude a prescribed algebraic degree. Its polynomial-relation proof is on pp. 14–17.
• Corollary 3, p. 5: irrationality with totient or divisor-sum exponents and small nonnegative weights. The proof on pp. 18–19 groups terms by exponent and uses two external inputs cited on p. 18: the Maier--Pomerance bound on the number of totient and divisor-sum values (Lemma 7) and a lower bound for the totient (Lemma 6, cited from Montgomery--Vaughan's book).

Theorem A on p. 2 restates Erdős's 1957 Lemma 4', whose omitted proof is supplied on p. 18 from Theorem 3. It and the sumset statement Corollary 1 on p. 4 remain unextracted here. The boxed coefficient size in the algebraic-base results is the maximum absolute value over conjugates, not the value in a single embedding; the result pages make that distinction explicit.

Reading coverage and relevance

The complete manuscript was read as extracted text. The five result statements, definitions and inherited gap condition were compared with the source; Theorem 3 and Corollaries 2–3 were also checked against their page images. This filing records statements, proof pointers and disclosed proof-route reading, not a complete source-proof reconstruction or independent review. No native tier is assigned. Full compilation and its independent review remain outstanding.

The original external proofs used in Corollary 3 were not read here: Montgomery--Vaughan's lower totient estimate and the Maier--Pomerance image-count estimate, with Ford cited for an improvement.

Bears on. Problem 249 as a possible sparse-series criterion. Direct application to its dense numerator sequence fails, and the corollary with totient exponents is a different series.

The pre-existing links to #68 and #247 are adjacent literature links. For problem 68 the paper is real but off-point: it treats sparse arithmetic-function series, not the irrationality of sum 1/(n!-1). For problem 247 it is likewise adjacent, documenting the active criterion literature without addressing transcendence of sum 1/2^{a_n}. This paper does not settle the factorial-denominator question in E0068 or the specific transcendence question in E0247 merely by providing these criteria. The general sparse-series method is the stated relationship.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.