On integer and fractional parts of powers of Salem numbers

Let N be a positive rational integer and let P be the set of powers of a Salem number of degree d. We prove that for any α∈P the fractional parts of the numbers , when n runs through the set of positive rational integers, are dense in the unit interval if and only if N≦ 2d − 4. We also show that for any α∈P the integer parts of the numbers αⁿ are divisible by N for infinitely many n if and only if N≦ 2d − 3. Received: 27 April 2005     

Arithmetical properties of linear recurrent sequences

Let F(z)∈R[z] be a polynomial with positive leading coefficient, and let α>1 be an algebraic number. For r=degF>0, assuming that at least one coefficient of F lies outside the field Q(α) if α is a Pisot number, we prove that the difference between the largest and the smallest limit points of the sequence of fractional parts {F(n)αn}_n=1,2,3,… is at least 1/?(Pr+1), where ? stands for the so-called reduced length of a polynomial.     

Integer parts of powers of rational numbers

We prove that the sequence [ξ(5/4)ⁿ], n=1,2, . . . , where ξ is an arbitrary positive number, contains infinitely many composite numbers. A corresponding result for the sequences [(3/2)ⁿ] and [(4/3)ⁿ],n=1,2, . . . , was obtained by Forman and Shapiro in 1967. Furthermore, it is shown that there are infinitely many positive integers n such that ([ξ(5/4)ⁿ],6006)>1, where 6006=2·3·7·11·13. Similar results are obtained for shifted powers of some other rational numbers. In particular, the same is proved for the sets of integers nearest to ξ(5/3)ⁿ and to ξ(7/5)ⁿ, n∈ℕ. The corresponding sets of possible divisors are also described.     

Salem numbers defined by Coxeter transformation

A real algebraic integer α>1 is called a Salem number if all its remaining conjugates have modulus at most 1 with at least one having modulus exactly 1. It is known [J.A. de la Peña, Coxeter transformations and the representation theory of algebras, in: V. Dlab et al. (Eds.), Finite Dimensional Algebras and Related Topics, Proceedings of the NATO Advanced Research Workshop on Representations of Algebras and Related Topics, Ottawa, Canada, Kluwer, August 10-18, 1992, pp. 223-253; J.F. McKee, P. Rowlinson, C.J. Smyth, Salem numbers and Pisot numbers from stars, Number theory in progress. in: K. Gy?ry et al. (Eds.), Proc. Int. Conf. Banach Int. Math. Center, Diophantine problems and polynomials, vol. 1, de Gruyter, Berlin, 1999, pp. 309-319; P. Lakatos, On Coxeter polynomials of wild stars, Linear Algebra Appl. 293 (1999) 159-170] that the spectral radii of Coxeter transformation defined by stars, which are neither of Dynkin nor of extended Dynkin type, are Salem numbers. We prove that the spectral radii of the Coxeter transformation of generalized stars are also Salem numbers. A generalized star is a connected graph without multiple edges and loops that has exactly one vertex of degree at least 3.     

On the distance from a rational power to the nearest integer

We prove that for any non-zero real number ξ the sequence of fractional parts {ξ(3/2)ⁿ}, n=1,2,3,…, contains at least one limit point in the interval [0.238117…,0.761882…] of length 0.523764…. More generally, it is shown that every sequence of distances to the nearest integer ||ξ(p/q)ⁿ||, n=1,2,3,…, where p/q>1 is a rational number, has both ‘large’ and ‘small’ limit points. All obtained constants are explicitly expressed in terms of p and q. They are also expressible in terms of the Thue-Morse sequence and, for irrational ξ, are best possible for every pair p>1, q=1. Furthermore, we strengthen a classical result of Pisot and Vijayaraghavan by giving similar effective results for any sequence ||ξαⁿ||, n=1,2,3,…, where α>1 is an algebraic number and where ξ≠0 is an arbitrary real number satisfying ξ∉Q(α) in case α is a Pisot or a Salem number.     

Several types of algebraic numbers on the unit circle

Let A _p ⊂ C denote the set of all algebraic numbers such that α ∈ A _p if and only if α is a zero of a (not necessarily irreducible) polynomial with positive rational coefficients. We give several results concerning the numbers in A _p . In particular, the intersection of A _p and the unit circle |z| = 1 is investigated in detail. So we determine all numbers of degree less than 6 on the unit circle which lie in the set A _p . Further we show that when α is a root of an irreducible rational polynomial p(X) of degree ≠ 4 whose Galois group contains the full alternating group, α lies in A _p if and only if no real root of p(X) is positive.Received: 19 November 2004; revised: 9 February 2005     

Littlewood Pisot numbers

A Pisot number is a real algebraic integer, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is a Littlewood polynomial, one with {+1,-1}-coefficients, and shows that they form an increasing sequence with limit 2. It is known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Littlewood polynomials. Finally, we prove that every reciprocal Littlewood polynomial of odd degree n?3 has at least three unimodular roots.     

Algebraic numbers and density modulo 1

Let λ₁,μ₁ and λ₂,μ₂ be two pairs of rationally independent real algebraic numbers of degree 2, with absolute values greater than 1, such that the absolute values of their conjugates are also greater than 1. Under some additional assumptions, on the relation between λi,μi and their conjugates, we prove that for any real numbers ξ₁,ξ₂, with at least one ξi≠0, the set is dense modulo 1.     

Heights and Tamagawa numbers of motives

K. Kato has recently defined and studied heights of mixed motives and proposed some interesting questions. In this article, we relate the study of heights to the study of Tamagawa numbers of motives. We also partially answer one of Kato's questions about the number of mixed motives of bounded heights in the case of mixed Tate motives with two graded quotients. Finally, we provide a concrete computation with the number of mixed Tate motives with three graded quotients.     

On Numbers which are Mahler Measures

We investigate which algebraic numbers can be Mahler measures. Adler and Marcus showed that these must be Perron numbers. We prove that certain integer multiples of every Perron number are Mahler measures. The results of Boyd give some necessary conditions on Perron number to be a measure. These do not include reciprocal algebraic integers, so it would be of interest to find one which is not a Mahler measure. We prove a result in this direction. Finally, we show that for every non-negative integer k there is a cubic algebraic integer having norm 2 such that precisely the kth iteration of its Mahler measure is an integer.     

On the compositum of integral closures of valuation rings

It is well known that if $K_1,K_2$ are algebraic number fields with coprime discriminants, then the composite ring $A_{K_1}{A}_{K_2}$ is integrally closed and $K_1,K_2$ are linearly disjoint over the field of rationals, $A_{K_i}$ being the ring of algebraic integers of $K_i$. In an attempt to prove the converse of the above result, in this paper we prove that if $K_1,K_2$ are finite separable extensions of a valued field $(K,v)$ of arbitrary rank which are linearly disjoint over $K=K_1\cap K_2$ and if the integral closure $S_i$ of the valuation ring $R_v$ of v in $K_i$ is a free $R_v$-module for $i=1,2$ with $S_1{S}_2$ integrally closed, then the discriminant of either $S_1/R_v$ or of $S_2/R_v$ is the unit ideal. We quickly deduce from this result that for algebraic number fields $K_1,K_2$ linearly disjoint over $K=K_1\cap K_2$ for which $A_{K_1}{A}_{K_2}$ is integrally closed, the relative discriminants of $K_1/K$ and $K_2/K$ must be coprime.     

Integer parts of powers of Pisot and Salem numbers

Let a \alpha be a Salem number or a Pisot number. We prove that integer parts of its powers [aⁿ] [\alpha^n] are composite for infinitely many integers n.     

Quadratic equations over finite fields and class numbers of real quadratic fields

Letp be an odd prime and the finite field withp elements. In the present paper we shall investigate the number of points of certain quadratic hypersurfaces in the vector space and derive explicit formulas for them. In addition, we shall show that the class number of the real quadratic field (wherep1 (mod 4)) over the field of rational numbers can be expressed by means of these formulas.     

On the transcendence of real numbers with a regular expansion

We apply the Ferenczi-Mauduit combinatorial condition obtained via a reformulation of Ridout's theorem to prove that a real number whose b-ary expansion is the coding of an irrational rotation on the circle with respect to a partition in two intervals is transcendental. We also prove the transcendence of real numbers whose b-ary expansion arises from a non-periodic three-interval exchange transformation.     

On Euler numbers modulo powers of two

To determine Euler numbers modulo powers of two seems to be a difficult task. In this paper we achieve this and apply the explicit congruence to give a new proof of a classical result due to M.A. Stern.     

Semilinear fractional elliptic equations involving measures

We study the existence of weak solutions to (E) (−Δ)^αu+g(u)=ν${(-\mathrm{\Delta })}^{\alpha }u+g(u)=\nu$ in a bounded regular domain Ω   in R^N(N≥2)${\mathbb{R}}^N(N\ge 2)$ which vanish in R^N?Ω${\mathbb{R}}^N?\Omega$, where (−Δ)^α${(-\mathrm{\Delta })}^{\alpha }$ denotes the fractional Laplacian with α∈(0,1)$\alpha \in (0,1)$, ν is a Radon measure and g is a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of weak solution for problem (E) for any measure. In the case where ν   is a Dirac measure, we characterize the asymptotic behavior of the solution. When g(r)=|r|^k−1r$g(r)={|r|}^{k-1}r$ with k supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.     

Indices of Kummer extensions of prime degree

Let p be a prime and let ${\zeta }_p$ be a primitive p-th root of unity. For a finite extension k of $\mathbb{Q}$ containing ${\zeta }_p$, we consider a Kummer extension $L/k$ of degree p. In this paper, we show that if $k=\mathbb{Q}({\zeta }_p)$ and the class number of k is one, the index of $L/k$ is one. We also show that if $L/k$ is tamely ramified with a normal integral basis, the index is at most a power of p. In the last section, we show that there exist infinitely many cubic Kummer extensions of $\mathbb{Q}({\zeta }_3)$ for both wildly and tamely ramified cases, whose integer rings do not have a power integral basis over that of $\mathbb{Q}({\zeta }_3)$.