Let $\alpha $ be an algebraic integer of degree $d$ with conjugates ${\alpha}_{1}=\alpha ,{\alpha}_{2},\cdots ,{\alpha}_{d}$. In the paper we give a lower bound for the mean value
$${M}_{p}\left(\alpha \right)=\sqrt[p]{\frac{1}{d}{\sum}_{i=1}^{d}|log|{\alpha}_{i}{\left|\right|}^{p}}$$
when $\alpha $ is not a root of unity and $p\>1$.

Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts ${a\u207f/n}_{n=1}^{\infty}$ 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.

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...

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.

We are interested whether there is a nonnegative integer ${u}_{0}$ and an infinite sequence of digits ${u}_{1},{u}_{2},{u}_{3},\cdots $ in base $b$ such that the numbers ${u}_{0}{b}^{n}+{u}_{1}{b}^{n-1}+\cdots +{u}_{n-1}b+{u}_{n},$ where $n=0,1,2,\cdots ,$ 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 $p\in S,$ then we call the set $S$ unavoidable with respect to $b$. It was proved earlier that unavoidable sets in base $b$ exist if $b\in \{2,3,4,6\},$ and that no unavoidable set exists in base $b=5.$ Now, we prove...

The main result of this paper implies that for every positive integer $d\u2a7e2$ 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.

We consider the sequence of fractional parts $\left\{\xi {\alpha}^{n}\right\}$, $n=1,2,3,\cdots $, where $\alpha >1$ is a Pisot number and $\xi \in \mathbb{Q}\left(\alpha \right)$ 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 $\xi =1$ and the unique limit point is zero, was earlier described by the author and Luca, independently.

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.

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...

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.

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 $1+{x}^{r\u2081}+\cdots +{x}^{r\u2085}$, 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...

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

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