Salem Numbers Research Articles

Norms extremal with respect to the Mahler measure

In this paper, we introduce and study several norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We show that bounding these norms on a certain subspace implies Lehmerʼs conjecture and in at least one case that the converse is true as well. We evaluate these norms on a class of algebraic numbers that include Pisot and Salem numbers, and for surds. We prove that the infimum in the construction is achieved in a certain finite dimensional space for all algebraic numbers in one case, and for surds in general, a finiteness result analogous to that of Samuels and Jankauskas for the t-metric Mahler measures.

THE TRACE PROBLEM FOR TOTALLY POSITIVE ALGEBRAIC INTEGERS

AbstractLetαbe a totally positive algebraic integer of degreed≥2 andα1=α,α2,…,αdbe all its conjugates. We use explicit auxiliary functions to improve the known lower bounds ofSk/d, whereSk=∑di=1αki andk=1,2,3. These improvements have consequences for the search of Salem numbers with negative traces.

On the minimum dilatation of pseudo-Anosov homeromorphisms on surfaces of small genus

We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our technique also simplifies Cho and Ham's proof of the least dilatation of pseudo-Anosov homeomorphisms on a genus two surface. For genus g=2 to 5, the mimimum dilatation is the smallest Salem number for polynomials of degree 2g.

Arithmetics in number systems with a negative base

We study the numeration system with a negative base, introduced by Ito and Sadahiro. We focus on arithmetic operations in the sets Fin ( − β ) and Z − β of numbers having finite resp. integer ( − β ) -expansions. We show that Fin ( − β ) is trivial if β is smaller than the golden ratio 1 2 ( 1 + 5 ) . For β ≥ 1 2 ( 1 + 5 ) we prove that Fin ( − β ) is a ring, only if β is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that Fin ( − β ) is a ring if β is a quadratic Pisot number with positive conjugate. For quadratic Pisot units, we determine the number of fractional digits that may appear when adding or multiplying two ( − β ) -integers.

Pisot Numbers from {0, 1}-Polynomials

AbstractA Pisot number is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer greater than 1, 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 derived from a Newman polynomial — one with {0, 1}-coefficients — and shows that they form a strictly increasing sequence with limit (1 + √5)/2. It has long been 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 Newman polynomials.

Galois theory of Salem polynomials

AbstractLet f(x) ∈ [x] be a monic irreducible reciprocal polynomial of degree 2d with roots r1, 1/r1, r2, 1/r2, . . ., rd, 1/rd. The corresponding trace polynomial g(x) of degree d is the polynomial whose roots are r1 + 1/r1, . . ., rd + 1/rd. If the Galois groups of f and g are Gf and Gg respectively, then Gg ≅ Gf/N, where N is isomorphic to a subgroup of C2d. In a naive sense, the generic case is Gf ≅ C2d ⋊ Sd, with N ≅ C2d and Gg ≅ Sd. When f(x) has extra structure this may be reflected in the Galois group, and it is not always true even that Gf ≅ N ⋊ Gg. For example, for cyclotomic polynomials f(x) = Φn(x) it is known that Gf ≅ N ⋊ Gg if and only if n is divisible either by 4 or by some prime congruent to 3 modulo 4.In this paper we deal with irreducible reciprocal monic polynomials f(x) ∈ [x] that are ‘close’ to being cyclotomic, in that there is one pair of real positive reciprocal roots and all other roots lie on the unit circle. With the further restriction that f(x) has degree at least 4, this means that f(x) is the minimal polynomial of a Salem number. We show that in this case one always has Gf ≅ N ⋊ Gg, and moreover that N ≅ C2d or C2d−1, with the latter only possible if d is odd.

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. Győry 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.

Equidistribution modulo 1 and Salem numbers

Let $\theta$ be a Salem number. It is well-known that the sequence $(\theta^n)$ modulo 1 is dense but not equidistributed. In this article we discuss equidistributed subsequences. Our first approach is computational and consists in estimating the supremum of $\lim_{n\to\infty} n/s(n)$ over all equidistributed subsequences $(\theta^{s(n)})$. As a result, we obtain an explicit upper bound on the density of any equidistributed subsequence. Our second approach is probabilistic. Defining a measure on the family of increasing integer sequences, we show that relatively to that measure, almost no subsequence is equiditributed.

Trace of totally positive algebraic integers and integer transfinite diameter

Explicit auxiliary functions can be used in the “Schur-Siegel- Smyth trace problem”. In the previous works, these functions were constructed only with polynomials having all their roots positive. Here, we use several polynomials with complex roots, which are found with Wu’s algorithm, and we improve the known lower bounds for the absolute trace of totally positive algebraic integers. This improvement has a consequence for the search of Salem numbers that have a negative trace. The same method also gives a small improvement of the upper bound for the integer transfinite diameter of [0,1].

Constructions of Pisot and Salem numbers with flat palindromes

This paper introduces explicit conditions for some natural family of polynomials to define Pisot or Salem numbers, and reviews related topics as well as their references.

Beta-expansions for infinite families of Pisot and Salem numbers

This paper continues the study of beta-expansions of 1 where β is a Pisot or Salem number. Special attention is given to regular Pisot numbers, and the Salem numbers that approach these Pisot numbers.

Monotonic sequences related to zeros of Bessel functions

In the course of their work on Salem numbers and uniform distribution modulo 1, A. Akiyama and Y. Tanigawa proved some inequalities concerning the values of the Bessel function J0 at multiples of π, i.e., at the zeros of J1/2. This raises the question of inequalities and monotonicity properties for the sequences of values of one cylinder function at the zeros of another such function. Here we derive such results by differential equations methods.

ON A FAMILY OF THREE TERM NONLINEAR INTEGER RECURRENCES

In the present paper we study sequences defined by the recurrence relation [Formula: see text] for n ≥ 0, where [Formula: see text] the golden ratio. These sequences are related to shift radix systems as well as to β-expansions with respect to Salem numbers.

On the lines passing through two conjugates of a Salem number

AbstractWe show that the number of distinct non-parallel lines passing through two conjugates of an algebraic number α of degree d ≥ 3 is at most [d2/2]-d+2, its conjugates being in general position if this number is attained. If, for instance, d ≥ 4 is even, then the conjugates of α ∈ $\overline{\Q}$ of degree d are in general position if and only if α has 2 real conjugates, d-2 complex conjugates, no three distinct conjugates of α lie on a line and any two lines that pass through two distinct conjugates of α are non-parallel, except for d/2-1 lines parallel to the imaginary axis. Our main result asserts that the conjugates of any Salem number are in general position. We also ask two natural questions about conjugates of Pisot numbers which lead to the equation α1+α2=α3+α4 in distinct conjugates of a Pisot number. The Pisot number $\al_1\,{=}\,(1+\sqrt{3+2\sqrt{5}})/2$ shows that this equation has such a solution.

Sumsets of Pisot and Salem numbers

Let S and T be the sets of Pisot and Salem numbers, respectively. We prove that the set mT ∩ T is empty for every positive integer m ⩾ 2 , i.e., that no sum of several Salem numbers is a Salem number. We also obtain a result which implies that the sets mT ∩ S and mS ∩ T are nonempty for every m ⩾ 2 , i.e., that certain Salem numbers can sum to a Pisot number and that certain Pisot numbers can sum to a Salem number. As an explicit example, the Salem number ( 22 + 10 5 + 5 10 5 + 14 + 50 5 + 70 ) / 8 = 10.99925 … is expressed by a sum of two Pisot numbers.

UNE MINORATION POUR L'EXPOSANT DU GROUPE DES CLASSES D'UN CORPS ENGENDRÉ PAR UN NOMBRE DE SALEM

In this article we extend the main result of [2] concerning lower bounds for the exponent of the class group of CM-fields. We consider a number field K generated by a Salem number α. If k denotes the field fixed by α ↦ α-1 we prove, under the generalized Riemann hypothesis for the Dedekind zeta function of K, lower bounds for the relative exponent eK/k and the relative size hK/k of the class group of K with respect to the class group of k.

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 \(\frac{{\alpha ^n }}{N}\), 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.

Remarks on a conjecture on certain integer sequences

The periodicity of sequences of integers ]]> ]]> ]]> ]]> ]]> ]]> ]]> (a_{n})_{n\in\mathbb Z}$ satisfying the inequalities ]]> 0 \le a_{n-1}+\lambda a_n +a_{n+1} < 1 (n \in {\mathbb Z}) $$ is studied for real $ \lambda $ with $|\lambda|< 2$. Periodicity is proved in case $ \lambda $ is the golden ratio; for other values of $ \lambda $ statements on possible period lengths are given. Further interesting results on the morphology of periods are illustrated. The problem is connected to the investigation of shift radix systems and of Salem numbers.

ARITHMETICAL PROPERTIES OF POWERS OF ALGEBRAIC NUMBERS

We consider the sequences of fractional parts The fact that there are infinitely many composite numbers in the sequence of integer parts of powers was proved earlier for Pisot numbers, Salem numbers and the three rational numbers 3/2, 4/3, 5/4, but no such algebraic number having several conjugates outside the unit circle was known.

ARITHMETICAL PROPERTIES OF POWERS OF ALGEBRAIC NUMBERS

We consider the sequences of fractional parts {ξαn}, n = 1, 2, 3,…, and of integer parts [ξαn], n = 1, 2, 3,…, where ξ is an arbitrary positive number and α > 1 is an algebraic number. We obtain an inequality for the difference between the largest and the smallest limit points of the first sequence. Such an inequality was earlier known for rational α only. It is also shown that for roots of some irreducible trinomials the sequence of integer parts contains infinitely many numbers divisible by either 2 or 3. This is proved, for instance, for [ ξ ( ( 13 − 1 ) / 2 ) n ] , n = 1, 2, 3,…. The fact that there are infinitely many composite numbers in the sequence of integer parts of powers was proved earlier for Pisot numbers, Salem numbers and the three rational numbers 3/2, 4/3, 5/4, but no such algebraic number having several conjugates outside the unit circle was known. 2000 Mathematics Subject Classification 11J71, 11R04, 11R06, 11A41.