Laursen 2024 transcendence certain sequences algebraic numbers

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Mathias L. Laursen, Transcendence of certain sequences of algebraic numbers. Research in Number Theory 10:70 (2024). arXiv:2308.15302, doi:10.1007/s40993-024-00553-2.

The paper extends the Erdos irrationality criterion for sequences (Theorem 1.1: a_n >= n^{1+eps} increasing with limsup a_n^{2^{-n}} = infinity implies sum 1/(a_n c_n) is irrational for all positive integer c_n) from integers to algebraic numbers lying in a fixed number field. Theorem 1.4 handles positive integers a_n with algebraic b_n written as integer combinations of fixed elements x_1,...,x_D of a number field K of degree d >= 2, giving irrationality of the sequence a_n/b_n when limsup a_n^{(dy/(1-beta)+1)^{-n}} = infinity and transcendence under the analogous condition with d^2 y; Theorem 1.6 shifts the arithmetic information back onto a_n at the cost of a more technical hypothesis. The engine is Schmidt's Subspace Theorem, which excludes nearly all algebraic numbers as values of the sum and leaves only values in a fixed number field, handled in the manner of Andersen-Kristensen. For sequences in one number field the results improve Andersen-Kristensen's Theorem 1.3 by allowing non-integral elements, weakening the condition that the house equal |a_n| and weakening the limsup thresholds. At d = 1 they give Corollary 7.1, which also follows from Hancl's Theorem 1.2; Section 7 notes that Theorem 1.2 is slightly stronger there and asks (Question 7.2) whether its larger b_n can be allowed. For problem 247 the paper is background rather than progress: all its criteria live in a doubly-exponential growth regime (limsup of a_n raised to a c^{-n} power), while the problem's question concerns limsup a_n/n = infinity for sum 1/2^{a_n}, which is left untouched here.

Source: https://arxiv.org/abs/2308.15302. The file prints "© The Author(s) 2024. This article is licensed under a Creative Commons Attribution 4.0 International License" on p. 1, with the license URL http://creativecommons.org/licenses/by/4.0/: the Creative Commons Attribution 4.0 license.

Bears on. #247

Results to transcribe.

• Theorem 1.1 (Erdos, quoted): If a_n is increasing with a_n >= n^{1+eps} and limsup a_n^{2^{-n}} = infinity, the sequence is irrational.
• Theorem 1.2 (Hancl, quoted): Transcendence criterion for ratios a_n/b_n of positive integers under a (3+gamma)^{-n} limsup condition.
• Theorem 1.3 (Andersen-Kristensen, quoted): Irrationality and transcendence for sequences of algebraic integers of bounded degree d, with limsup exponents built from products of (d^i+d)^{-1}.
• Theorem 1.4: For positive integers n^{1+eps} <= a_n <= a_{n+1}, and non-zero b_n = sum b_{i,n} x_i with b_{i,n} integers and x_1,...,x_D fixed in a number field K of degree d >= 2, where for large n |b_n| <= a_n^beta 2^{log_2^alpha a_n} with 0 <= beta < eps/(1+eps), |b_{i,n}| <= a_n^y 2^{log_2^alpha a_n} with 0 < alpha < 1 <= y, and Re(zeta b_n) > 0 for a fixed complex zeta, the sequence a_n/b_n is irrational if limsup a_n^{(dy/(1-beta)+1)^{-n}} = infinity and transcendental with d^2 y in place of dy.
• Theorem 1.6: Variant of Theorem 1.4 in which a_n itself is an integer combination of the fixed elements x_1,...,x_D of K and b_n is a positive integer, so the arithmetic information sits in a_n.
• Remark 1.5: The positivity hypothesis (Re(zeta b_n) > 0) is used only to keep the partial sums from repeating a value infinitely often and can be replaced by any hypothesis with that effect.