On the fractional parts of the natural powers of a fixed number.
Infinite sets of integers whose distinct elements do not sum to a power.
Periodicity of some recurrence sequences modulo .
The mean values of logarithms of algebraic integers
Let be an algebraic integer of degree with conjugates . In the paper we give a lower bound for the mean value when is not a root of unity and .
On a conjecture of A. Schinzel and H. Zassenhaus
On intervals containing full sets of conjugates of algebraic integers
Three problems for polynomials of small measure
Additive relations with conjugate algebraic numbers
Powers of a rational number modulo 1 cannot lie in a small interval
Density of some sequences modulo 1
Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts 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 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
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 be the mth coefficient of the square f(x)² of a unimodular...
Squares and cubes in Sturmian sequences
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.
Mahler measures in a cubic field
We prove that every cyclic cubic extension of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in . This extends the result of Schinzel who proved the same statement for every real quadratic field . A corresponding conjecture is made for an arbitrary non-totally complex field 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
We are interested whether there is a nonnegative integer and an infinite sequence of digits in base such that the numbers where are all prime or at least do not have prime divisors in a finite set of prime numbers If any such sequence contains infinitely many elements divisible by at least one prime number then we call the set unavoidable with respect to . It was proved earlier that unavoidable sets in base exist if and that no unavoidable set exists in base Now, we prove...
On the limit points of the fractional parts of powers of Pisot numbers
We consider the sequence of fractional parts , , where is a Pisot number and 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 and the unique limit point is zero, was earlier described by the author and Luca, independently.
Nonreciprocal algebraic numbers of small measure
The main result of this paper implies that for every positive integer there are at least nonconjugate algebraic numbers which have their Mahler measures lying in the interval . These algebraic numbers are constructed as roots of certain nonreciprocal quadrinomials.
Two Exercises Concerning the Degree of the Product of Algebraic Numbers
On polynomials with flat squares
Sumsets without powerful numbers
On degrees of three algebraic numbers with zero sum or unit product
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...
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