On semi-strong U-numbers
On Siegel's Lemma.
On Siegel-Shidlovskii's theory for q-difference equations
On simultaneous inhomogeneous Diophantine approximation
On simultaneous rational approximation to a real number and its integral powers
For a positive integer and a real number , let denote the supremum of the real numbers such that there are arbitrarily large positive integers such that are all less than . Here, denotes the distance to the nearest integer. We study the set of values taken by the function and, more generally, we are concerned with the joint spectrum of . We further address several open problems.
On simultaneous rational approximations to a real number, its square, and its cube
On some arithmetical properties of weighted sums of S-units.
On some consequences of Thue's principle.
On some non-holonomic sequences.
On some special forms of simultaneous Pell equations
On Sprindzuk's classification of transcendental numbers.
On subsequences of convergents to a quadratic irrational given by some numerical schemes
Given a quadratic irrational , we are interested in how some numerical schemes applied to a convenient function provide subsequences of convergents to . We investigate three numerical schemes: secant-like methods and formal generalizations, which lead to linear recurring subsequences; the false position method, which leads to arithmetical subsequences of convergents and gives some interesting series expansions; Newton’s method, for which we complete a result of Edward Burger [1] about the existence...
On substitution invariant Sturmian words: an application of Rauzy fractals
Sturmian words are infinite words that have exactly n+1 factors of length n for every positive integer n. A Sturmian word sα,p is also defined as a coding over a two-letter alphabet of the orbit of point ρ under the action of the irrational rotation Rα : x → x + α (mod 1). A substitution fixes a Sturmian word if and only if it is invertible. The main object of the present paper is to investigate Rauzy fractals associated with two-letter invertible substitutions. As an application, we give...
On sums of -units and linear recurrences
On systems of linear inequalities
We show in detail that the category of general Roth systems or the category of semi-stable systems of linear inequalities of slope zero is a neutral Tannakian category. On the way, we present a new proof of the semi-stability of the tensor product of semi-stable systems. The proof is based on a numerical criterion for a system of linear inequalities to be semi-stable.
On terms of linear recurrence sequences with only one distinct block of digits
In 2000, Florian Luca proved that F₁₀ = 55 and L₅ = 11 are the largest numbers with only one distinct digit in the Fibonacci and Lucas sequences, respectively. In this paper, we find terms of a linear recurrence sequence with only one block of digits in its expansion in base g ≥ 2. As an application, we generalize Luca's result by finding the Fibonacci and Lucas numbers with only one distinct block of digits of length up to 10 in its decimal expansion.
On the abc conjecture.
On the abc conjecture in algebraic number fields
On the addition of residue classes mod p
On the Algebraic Independence of Holomorphic Solutions of Certain Functional Equations and their Values.