Salem Numbers Research Articles

An arithmetical property of powers of Salem numbers

Let ζ be a nonzero real number and let α be a Salem number. We show that the difference between the largest and smallest limit points of the fractional parts of the numbers ζ α n , when n runs through the set of positive rational integers, can be bounded below by a positive constant depending only on α if and only if the algebraic integer α − 1 is a unit.

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 | | ξ α n | | , 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.

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.

THERE ARE SALEM NUMBERS OF EVERY TRACE

We show that there are Salem numbers of every trace. The nontrivial part of this result is for Salem numbers of negative trace. The proof has two main ingredients. The first is a novel construction, using pairs of polynomials whose zeros interlace on the unit circle, of polynomials of specified negative trace having one factor a Salem polynomial, with any other factors being cyclotomic. The second is an upper bound for the exponent of a maximal torsion coset of an algebraic torus in a variety defined over the rationals. This second result, which may be of independent interest, enables us to refine our construction to avoid getting cyclotomic factors, giving a Salem polynomial of any specified trace, with a trace-dependent bound for its degree. We show also how our interlacing construction can be easily adapted to produce Pisot polynomials, giving a simpler, and more explicit, construction for Pisot numbers of arbitrary trace than previously known.

Salem numbers and uniform distribution modulo 1

For a Salem number fi of degree d, the distridution of fractional parts of fi n (n = 1;2;:::) is studied. By giving explicit inequalities, it is shown to be 'exponentially' close to uniform distribution when d is large.

Transcendance « à la Liouville » de certains nombres réels

Using a Liouville-like approach, we prove that real numbers like ∑ n=0 +∞(1/ β u n ), where β is a Pisot or a Salem number and where ( u n ) n⩾0 is a sufficiently lacunary sequence of positive integers, are transcendental. To cite this article: B. Adamczewski, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

ε-Pisot numbers in any real algebraic number field are relatively dense

An algebraic integer is called an ε-Pisot number ( ε>0) if its Galois conjugates have absolute value less then ε. Let K be any real algebraic number field. We prove that the subset of K consisting of ε-Pisot numbers which have the same degree as that of the field is relatively dense in the real line R . This has some applications to non-stationary products of random matrices involving Salem numbers.

The Schur algorithm for generalized Schur functions III: J-unitary matrix polynomials on the circle

The main result is that for J= 1 0 0 −1 every J-unitary 2×2-matrix polynomial on the unit circle is an essentially unique product of elementary J-unitary 2×2-matrix polynomials which are either of degree 1 or 2 k. This is shown by means of the generalized Schur transformation introduced in [Ann. Inst. Fourier 8 (1958) 211; Ann. Acad. Sci. Fenn. Ser. A I 250 (9) (1958) 1–7] and studied in [Pisot and Salem Numbers, Birkhäuser Verlag, Basel, 1992; Philips J. Res. 41 (1) (1986) 1–54], and also in the first two parts [Operator Theory: Adv. Appl. 129, Birkhäuser Verlag, Basel, 2000, p. 1; Monatshefte für Mathematik, in press] of this series. The essential tool in this paper are the reproducing kernel Pontryagin spaces associated with generalized Schur functions.

Automorphisms of even unimodular lattices and unramified Salem numbers

In this paper we study the characteristic polynomials S ( x )=det( xI − F |II p , q ) of automorphisms of even unimodular lattices with signature ( p , q ). In particular, we show that any Salem polynomial of degree 2 n satisfying S (−1) S (1)=(−1) n arises from an automorphism of an indefinite lattice, a result with applications to K3 surfaces.

Integer parts of powers of Pisot and Salem numbers

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

Dynamics on K3 surfaces: Salem numbers and Siegel disks

This paper presents the first examples of K3 surface automorphisms \(f : X \rightarrow X\) with Siegel disks (domains on which f acts by an irrational rotation). The set of such examples is countable, and the surface \(X\) must be non-projective to carry a Siegel disk. These automorphisms are synthesized from Salem numbers of degree 22 and trace −1, which play the role of the leading eigenvalue for \(f*|H^2(X)\). The construction uses the Torelli theorem, the Atiyah-Bott fixed-point theorem and results from transcendence theory.

Some computations on the spectra of Pisot and Salem numbers

Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdos. Joo and Komornik in 1990. is the determination of l(q) for Pisot numbers q, where l(q) = inf(|y|:y = e0 + e1q1 + ...+ enqn, ei ∈ {±1,0}, y ≠ 0.) Although the quantity l(q) is known for some Pisot numbers q, there has been no general method for computing l(q). This paper gives such an algorithm. With this algorithm, some properties of l(q) and its generalizations are investigated.A related question concerns the analogy of l(q), denoted a(q), where the coefficients are restricted to ±1; in particular, for which non-Pisot numbers is a(q) nonzero? This paper finds an infinite class of Salem numbers where a(q) ≠ 0.

The arithmetic and geometry of Salem numbers

A Salem number is a real algebraic integer, greater than 1 1 , with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and arithmetic, and discuss the possible implications for the ‘minimization problem’. This is an old question in number theory which asks whether the set of Salem numbers is bounded away from 1 1 .

ON THE REMAK HEIGHT, THE MAHLER MEASURE AND CONJUGATE SETS OF ALGEBRAIC NUMBERS LYING ON TWO CIRCLES

AbstractWe define a new height function $\mathcal{R}(\alpha)$, the Remak height of an algebraic number $\alpha$. We give sharp upper and lower bounds for $\mathcal{R}(\alpha)$ in terms of the classical Mahler measure $M(\alpha)$. Study of when one of these bounds is exact leads us to consideration of conjugate sets of algebraic numbers of norm $\pm 1$ lying on two circles centred at 0. We give a complete characterization of such conjugate sets. They turn out to be of two types: one related to certain cubic algebraic numbers, and the other related to a non-integer generalization of Salem numbers which we call extended Salem numbers.AMS 2000 Mathematics subject classification: Primary 11R06

ON THE TRACE OF ALGEBRAIC INTEGERS OF SMALL HEIGHT

We give an upper bound for the modulus of the first non–zero trace among natural powers of an algebraic integer of small house. An upper bound for this power is obtained for the Pisot and Salem numbers. Although the house of these numbers is not at all small, similar bounds for the first non–zero trace are also established. Finally, we give an upper bound for the trace of an algebraic number with the Mahler measure bounded above by the square root of the degree.

Salem numbers of negative trace

We prove that, for all d > 4, there are Salem numbers of degree 2d and trace - 1, and that the number of such Salem numbers is » d/ (log log d) 2 . As a consequence, it follows that the number of totally positive algebraic integers of degree d and trace 2d - 1 is also » d/ (log log d) 2 .

There are Only Eleven Special Pisot Numbers

A special Pisot number is a Pisot number α such that α/(α−1) is also a Pisot number. Lagarias, Porta and Stolarsky found 11 special Pisot numbers. Here it is proved that there are no more. Also, the one known Salem number τ with the property that τ/(τ−1) is a Pisot number is shown to be the only such number.

Polynomials with small Mahler measure

We describe several searches for polynomials with integer coefficients and small Mahler measure. We describe the algorithm used to test Mahler measures. We determine all polynomials with degree at most 24 and Mahler measure less than 1.3 1.3 , test all reciprocal and antireciprocal polynomials with height 1 and degree at most 40, and check certain sparse polynomials with height 1 and degree as large as 181. We find a new limit point of Mahler measures near 1.309 1.309 , four new Salem numbers less than 1.3 1.3 , and many new polynomials with small Mahler measure. None has measure smaller than that of Lehmer’s degree 10 polynomial.

Pisot and Salem numbers, by M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, J. P. Schreiber. Pp. 291. SuFr 158. 1992. ISBN 3-7543-2548-4 (Birkhäuser).

Pisot and Salem numbers, by M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, J. P. Schreiber. Pp. 291. SuFr 158. 1992. ISBN 3-7543-2548-4 (Birkhäuser).

Harmonious orbits of linear transformations

We describe necessary and sufficient conditions for orbits of linear transformations onR n ,n≥1, and sets arising as sums of elements from orbits, to be harmonious subsets. This is done via a generalization of the notion of Pisot-Vijayaraghavan and Salem numbers.