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1
Content available remote

Three problems for polynomials of small measure

100%
Dubickas A.
Acta Arithmetica
|
2001
|
tom 98
|
nr 3
279-292
2
Content available remote

Density of some sequences modulo 1

100%
Dubickas A.
Colloquium Mathematicum
|
2012
|
tom 128
|
nr 2
237-244
EN
Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts ${aⁿ/n}_{n=1}^{∞}$ 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 $cN^{-0.475}$ 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.
3
Content available remote

On a conjecture of A. Schinzel and H. Zassenhaus

100%
Dubickas A.
Acta Arithmetica
|
1993
|
tom 63
|
nr 1
15-20
4
Content available remote

On intervals containing full sets of conjugates of algebraic integers

100%
Dubickas A.
Acta Arithmetica
|
1999
|
tom 91
|
nr 4
379-386
5
Content available remote

Heights of squares of Littlewood polynomials and infinite series

100%
Dubickas A.
Annales Polonici Mathematici
|
2012
|
tom 105
|
nr 2
145-153
EN
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 series $f(x) = ∑_{i=0}^{∞} a_i x^i$, where all $a_i ∈ ℂ$ satisfy $|a_i| = 1$. We show that then $lim sup_{m → ∞} |A_m|/√m ≥ 1$ and that there exist some infinite series with ±1 coefficients and an integer m(ε) such that $|A_m| < (2+ε)√(mlogm)$ for each m ≥ m(ε).
6
Content available remote

Additive relations with conjugate algebraic numbers

100%
Dubickas A.
Acta Arithmetica
|
2003
|
tom 107
|
nr 1
35-43
7
Content available remote

Powers of a rational number modulo 1 cannot lie in a small interval

100%
Dubickas A.
Acta Arithmetica
|
2009
|
tom 137
|
nr 3
233-239
8
Content available remote

Auxiliary polynomials for some problems regarding Mahler's measure

64%
Dubickas A., Mossinghoff M.
Acta Arithmetica
|
2005
|
tom 119
|
nr 1
65-79
9
Content available remote

On polynomials with flat squares

64%
Dubickas A., Šemetulskis G.
Acta Arithmetica
|
2011
|
tom 146
|
nr 3
247-255
10
Content available remote

Sumsets without powerful numbers

64%
Dubickas A., Stankevičius A.
Acta Arithmetica
|
2007
|
tom 130
|
nr 4
381-387
11
Content available remote

Multiplicative dependence of shifted algebraic numbers

64%
Drungilas P., Dubickas A.
Colloquium Mathematicum
|
2003
|
tom 96
|
nr 1
75-81
EN
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.
12
Content available remote

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

64%
Drungilas P., Dubickas A.
Colloquium Mathematicum
|
2016
|
tom 143
|
nr 2
159-167
EN
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 compositum KL over k equals c. Towards another earlier formulated conjecture, under certain natural assumptions (related to the inverse Galois problem), we show that the set of such triplets forms a multiplicative semigroup.
13
Content available remote

Nonreciprocal algebraic numbers of small Mahler's measure

64%
Dubickas A., Jankauskas J.
Acta Arithmetica
|
2013
|
tom 157
|
nr 4
357-364
EN
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 $1 + x^{r₁} + ⋯ + x^{r₅}$, where the integers 1 ≤ r₁ < ⋯ < r₅ ≤ d satisfy some restrictions including $2r_{j} < r_{j+1}$ for j = 1,2,3,4. This result improves the previous lower bound cd³ and seems to be closer to the correct value of this function in d than the best known upper bound which is exponential in d.
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