scholarly journals Diophantine Approximations of Infinite Series and Products

2016 ◽  
Vol 24 (1) ◽  
pp. 71-82 ◽  
Author(s):  
Ondřej Kolouch ◽  
Lukáš Novotný
Keyword(s):  
Infinite Series ◽  
Linear Independence ◽  
Infinite Products ◽  
Survey Paper ◽  
Diophantine Approximations

Abstract This survey paper presents some old and new results in Diophantine approximations. Some of these results improve Erdos’ results on irrationality. The results in irrationality, transcendence and linear independence of infinite series and infinite products are put together with idea of irrational sequences and expressible sets.

On Diophantine approximations of the solutions of q-functional equations

2002 ◽  
Vol 132 (3) ◽  
pp. 639-659 ◽  
Author(s):  
Tapani Matala-aho
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Linear Independence ◽  
Number Fields ◽  
Growth Conditions ◽  
Linear Forms ◽  
Dimension Estimate ◽  
Linearly Independent ◽  
Diophantine Approximations ◽  
Image Position

Given a sequence of linear forms in m ≥ 2 complex or p-adic numbers α1, …,αm ∈ Kv with appropriate growth conditions, Nesterenko proved a lower bound for the dimension d of the vector space Kα1 + ··· + Kαm over K, when K = Q and v is the infinite place. We shall generalize Nesterenko's dimension estimate over number fields K with appropriate places v, if the lower bound condition for |Rn| is replaced by the determinant condition. For the q-series approximations also a linear independence measure is given for the d linearly independent numbers. As an application we prove that the initial values F(t), F(qt), …, F(qm−1t) of the linear homogeneous q-functional equation where N = N(q, t), Pi = Pi(q, t) ∈ K[q, t] (i = 1, …, m), generate a vector space of dimension d ≥ 2 over K under some conditions for the coefficient polynomials, the solution F(t) and t, q ∈ K*.

scholarly journals A Criterion for linear independence of infinite products

2015 ◽  
Vol 23 (2) ◽  
pp. 107-120
Author(s):  
Jaroslav Hančl ◽  
Ondřej Kolouch ◽  
Lukáš Novotný
Keyword(s):  
Linear Independence ◽  
Rational Numbers ◽  
Infinite Products

Abstract Using an idea of Erdős the paper establishes a criterion for the linear independence of infinite products which consist of rational numbers. A criterion for irrationality is obtained as a consequence.

On Linear Independence of a Certain Multivariate Infinite Product

2008 ◽  
Vol 51 (1) ◽  
pp. 32-46 ◽  
Author(s):  
Stephen Choi ◽  
Ping Zhou
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Linear Independence ◽  
Infinite Product ◽  
Infinite Products ◽  
Image Position ◽  
Positive Integers

AbstractLet q,m,M ≥ 2 be positive integers and r1, r2, … , rm be positive rationals and consider the following M multivariate infinite productsfor i = 0, 1, … ,M –1. In this article, we study the linear independence of these infinite products. In particular, we obtain a lower bound for the dimension of the vector space ℚF0+ℚF1+· · ·+ℚFM–1+ℚ over ℚ and show that among these M infinite products, F0, F1, … , FM–1, at least ∼ M/m(m + 1) of them are irrational for fixed m and M → ∞.

Refinement of the Chowla–Erdős method and linear independence of certain Lambert series

2019 ◽  
Vol 31 (6) ◽  
pp. 1557-1566
Author(s):  
Daniel Duverney ◽  
Yohei Tachiya
Keyword(s):  
Infinite Series ◽  
Rational Number ◽  
Linear Independence ◽  
Arithmetic Function ◽  
Growth Conditions ◽  
Necessary Condition ◽  
Lambert Series ◽  
Independence Results

AbstractIn this paper, we refine the method of Chowla and Erdős on the irrationality of Lambert series and study a necessary condition for the infinite series {\sum\theta(n)/q^{n}} to be a rational number, where q is an integer with {|q|>1} and θ is an arithmetic function with suitable divisibility and growth conditions. As applications of our main theorem, we give linear independence results for various kinds of Lambert series.

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