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The mean values of logarithms of algebraic integers

Artūras Dubickas — 1998

Journal de théorie des nombres de Bordeaux

Let α be an algebraic integer of degree d with conjugates α 1 = α , α 2 , , α d . In the paper we give a lower bound for the mean value M p ( α ) = 1 d i = 1 d | log | α i | | p p when α is not a root of unity and p > 1 .

Density of some sequences modulo 1

Artūras Dubickas — 2012

Colloquium Mathematicae

Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts a / n n = 1 is everywhere dense in the interval [0,1]. We prove a similar result for all Pisot numbers and Salem numbers α and show that for each c > 0 and each sufficiently large N, every subinterval of [0,1] of length c N - 0 . 475 contains at least one fractional part Q(αⁿ)/n, where Q is a nonconstant polynomial in ℤ[z] and n is an integer satisfying 1 ≤ n ≤ N.

Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas — 2012

Annales Polonici Mathematici

Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let A m be the mth coefficient of the square f(x)² of a unimodular...

Mahler measures in a cubic field

Artūras Dubickas — 2006

Czechoslovak Mathematical Journal

We prove that every cyclic cubic extension E of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in E . This extends the result of Schinzel who proved the same statement for every real quadratic field E . A corresponding conjecture is made for an arbitrary non-totally complex field E and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure.

Truncatable primes and unavoidable sets of divisors

Artūras Dubickas — 2006

Acta Mathematica Universitatis Ostraviensis

We are interested whether there is a nonnegative integer u 0 and an infinite sequence of digits u 1 , u 2 , u 3 , in base b such that the numbers u 0 b n + u 1 b n - 1 + + u n - 1 b + u n , where n = 0 , 1 , 2 , , are all prime or at least do not have prime divisors in a finite set of prime numbers S . If any such sequence contains infinitely many elements divisible by at least one prime number p S , then we call the set S unavoidable with respect to b . It was proved earlier that unavoidable sets in base b exist if b { 2 , 3 , 4 , 6 } , and that no unavoidable set exists in base b = 5 . Now, we prove...

Nonreciprocal algebraic numbers of small measure

Artūras Dubickas — 2004

Commentationes Mathematicae Universitatis Carolinae

The main result of this paper implies that for every positive integer d 2 there are at least ( d - 3 ) 2 / 2 nonconjugate algebraic numbers which have their Mahler measures lying in the interval ( 1 , 2 ) . These algebraic numbers are constructed as roots of certain nonreciprocal quadrinomials.

On the limit points of the fractional parts of powers of Pisot numbers

Artūras Dubickas — 2006

Archivum Mathematicum

We consider the sequence of fractional parts { ξ α n } , n = 1 , 2 , 3 , , where α > 1 is a Pisot number and ξ ( α ) is a positive number. We find the set of limit points of this sequence and describe all cases when it has a unique limit point. The case, where ξ = 1 and the unique limit point is zero, was earlier described by the author and Luca, independently.

Squares and cubes in Sturmian sequences

Artūras Dubickas — 2009

RAIRO - Theoretical Informatics and Applications

We prove that every Sturmian word has infinitely many prefixes of the form , where and lim In passing, we give a very simple proof of the known fact that every Sturmian word begins in arbitrarily long squares.

On degrees of three algebraic numbers with zero sum or unit product

Paulius DrungilasArtūras Dubickas — 2016

Colloquium Mathematicae

Let α, β and γ be algebraic numbers of respective degrees a, b and c over ℚ such that α + β + γ = 0. We prove that there exist algebraic numbers α₁, β₁ and γ₁ of the same respective degrees a, b and c over ℚ such that α₁ β₁ γ₁ = 1. This proves a previously formulated conjecture. We also investigate the problem of describing the set of triplets (a,b,c) ∈ ℕ³ for which there exist finite field extensions K/k and L/k (of a fixed field k) of degrees a and b, respectively, such that the degree of the...

Multiplicative dependence of shifted algebraic numbers

Paulius DrungilasArtūras Dubickas — 2003

Colloquium Mathematicae

We show that the set obtained by adding all sufficiently large integers to a fixed quadratic algebraic number is multiplicatively dependent. So also is the set obtained by adding rational numbers to a fixed cubic algebraic number. Similar questions for algebraic numbers of higher degrees are also raised. These are related to the Prouhet-Tarry-Escott type problems and can be applied to the zero-distribution and universality of some zeta-functions.

Nonreciprocal algebraic numbers of small Mahler's measure

Artūras DubickasJonas Jankauskas — 2013

Acta Arithmetica

We prove that there exist at least cd⁵ monic irreducible nonreciprocal polynomials with integer coefficients of degree at most d whose Mahler measures are smaller than 2, where c is some absolute positive constant. These polynomials are constructed as nonreciprocal divisors of some Newman hexanomials 1 + x r + + x r , where the integers 1 ≤ r₁ < ⋯ < r₅ ≤ d satisfy some restrictions including 2 r j < r j + 1 for j = 1,2,3,4. This result improves the previous lower bound cd³ and seems to be closer to the correct value of...

Linear recurrence sequences without zeros

Artūras DubickasAivaras Novikas — 2014

Czechoslovak Mathematical Journal

Let a d - 1 , , a 0 , where d and a 0 0 , and let X = ( x n ) n = 1 be a sequence of integers given by the linear recurrence x n + d = a d - 1 x n + d - 1 + + a 0 x n for n = 1 , 2 , 3 , . We show that there are a prime number p and d integers x 1 , , x d such that no element of the sequence X = ( x n ) n = 1 defined by the above linear recurrence is divisible by p . Furthermore, for any nonnegative integer s there is a prime number p 3 and d integers x 1 , , x d such that every element of the sequence X = ( x n ) n = 1 defined as above modulo p belongs to the set { s + 1 , s + 2 , , p - s - 1 } .

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