Salem Numbers Research Articles

Multiplicities in the Length Spectrum and Growth Rate of Salem Numbers

Multiplicities in the Length Spectrum and Growth Rate of Salem Numbers

There are Salem Numbers with Trace –3 and Every Degree At Least 34

We prove that there exist Salem numbers with trace –3 and every even degree ≥ 34 . Our proof combines a theoretical approach, which allows us to treat all sufficiently large degrees, with a numerical search for small degrees. Since it is known that there are no Salem numbers of trace –3 and degree ≤ 30 , our result is optimal up to possibly the single value 32, for which it is expected there are no such numbers.

On Sequences of Zeros of Polynomials Involving Pisot and Salem Numbers

On Sequences of Zeros of Polynomials Involving Pisot and Salem Numbers

Constructing pseudo-Anosovs from expanding interval maps

We investigate a phenomenon observed by Thurston wherein one constructs a pseudo-Anosov homeomorphism on the limit set of a certain lift of a piecewise linear expanding interval map. We reconcile this construction with a special subclass of generalized pseudo-Anosovs, first defined by de Carvalho. From there we classify the circumstances under which this construction produces a pseudo-Anosov. As an application, we produce for each g \geq 1 , a pseudo-Anosov \phi_g on the closed surface of genus g that preserves an algebraically primitive translation structure and whose dilatation is a Salem number.

On periodic alternate base expansions

For an alternate base β=(β0,…,βp−1), we show that if all rational numbers in the unit interval [0,1) have periodic expansions with respect to the p shifts of β, then the bases β0,…,βp−1 all belong to the extension field Q(β) where β is the product β0⋯βp−1 and moreover, this product β must be either a Pisot number or a Salem number. We also prove the stronger statement that if the bases β0,…,βp−1 belong to Q(β) but the product β is neither a Pisot number nor a Salem number then the set of rationals having an ultimately periodic β-expansion is nowhere dense in [0,1). Moreover, in the case where the product β is a Pisot number and the bases β0,…,βp−1 all belong to Q(β), we prove that the set of points in [0,1) having an ultimately periodic β-expansion is precisely the set Q(β)∩[0,1). For the restricted case of Rényi real bases, i.e., for p=1 in our setting, our method gives rise to an elementary proof of Schmidt's original result. Therefore, even though our results generalize those of Schmidt, our proofs should not be seen as generalizations of Schmidt's original arguments but as an original method in the generalized framework of alternate bases, which moreover gives a new elementary proof of Schmidt's results from 1980. As an application of our results, we show that if β=(β0,…,βp−1) is an alternate base such that the product β of the bases is a Pisot number and β0,…,βp−1∈Q(β), then β is a Parry alternate base, meaning that the quasi-greedy expansions of 1 with respect to the p shifts of the base β are ultimately periodic.

Every Salem number is a difference of two Pisot numbers

AbstractIn this note, we prove that every Salem number is expressible as a difference of two Pisot numbers. More precisely, we show that for each Salem number α of degree d, there are infinitely many positive integers n for which $\alpha^{2n-1}-\alpha^n+\alpha$ and $\alpha^{2n-1}-\alpha^n$ are both Pisot numbers of degree d and that the smallest such n is at most $6^{d/2-1}+1$. We also prove that every real positive algebraic number can be expressed as a quotient of two Pisot numbers. Earlier, Salem himself had proved that every Salem number can be written in this way.

Lattice isometries and K3 surface automorphisms: Salem numbers of degree 20

This article extends Bayer-Fluckiger's theorem on characteristic polynomials of isometries on an even unimodular lattice to the case where the isometries have determinant −1. As an application, we show that the logarithm of every Salem number of degree 20 is realized as the topological entropy of an automorphism of a nonprojective K3 surface.

Bottom of the length spectrum of arithmetic orbifolds

We prove that cocompact arithmetic lattices in a simple Lie group are uniformly discrete if and only if Salem numbers are uniformly bounded away from 1 1 . We also prove an analogous result for semisimple Lie groups. Finally, we shed some light on the structure of the bottom of the length spectrum of an arithmetic orbifold Γ ∖ X \Gamma \backslash X by showing the existence of a positive constant δ ( X ) > 0 \delta (X)>0 such that squares of lengths of closed geodesics shorter than δ \delta must be pairwise linearly dependent over Q \mathbb {Q} .

Salem numbers with minimal trace

In this paper, a new method to compute lower and upper bounds for Salem numbers with a given trace and a given degree is given. With this method, it is proven that the smallest trace of Salem numbers of degree 22 22 is − 1 -1 . Further, new lower bounds for degree of Salem numbers with minimal trace − 5 -5 and − 6 -6 are given. All Salem numbers of trace − 2 -2 and degree 24 24 , 26 26 are given. This includes 7 7 additional Salem numbers of degree 26 26 beyond what was previously known. The auxiliary functions related to Chebyshev polynomials, which are adapted to Salem number, are used in this work.

K3 surfaces, Picard numbers and Siegel disks

If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between 0 and 18. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type fixed point formulas and related Grothendieck residues.

Periodic expansion of one by Salem numbers

AbstractWe show that for a Salem number $\beta $ of degree d, there exists a positive constant $c(d)$ where $\beta ^m$ is a Parry number for integers m of natural density $\ge c(d)$ . Further, we show $c(6)>1/2$ and discuss a relation to the discretized rotation in dimension $4$ .

On a Class of Lacunary Almost Newman Polynomials Modulo P and Density Theorems

Abstract The reduction modulo p of a family of lacunary integer polynomials, associated with the dynamical zeta function ζβ (z)of the β-shift, for β> 1 close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in 𝔽 p and their factorizations, via Kronecker’s Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in 𝔽 p is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.

The digit exchanges in the rotational beta expansions of algebraic numbers

In this article, we investigate the β-expansions of real algebraic numbers. In particular, we give new lower bounds for the number of digit exchanges in the case where β is a Pisot or Salem number. Moreover, we define a new class of algebraic numbers, quasi-Pisot numbers and quasi-Salem numbers, which gives a generalization of Pisot numbers and Salem numbers.Our method is applicable also to the digit expansions of complex algebraic numbers, which gives a new estimate. In particular, we investigate the digits of rotational beta expansion considered by Akiyama and Caalim [3] and zeta-expansion by Surer [20], where the base is a quasi-Pisot or quasi-Salem number.

SALEM NUMBERS, SPECTRAL RADII AND GROWTH RATES OF HYPERBOLIC COXETER GROUPS

We show that not every Salem number appears as the growth rate of a cocompact hyperbolic Coxeter group. We also give a new proof of the fact that the growth rates of planar hyperbolic Coxeter groups are spectral radii of Coxeter transformations, and show that this need not be the case for growth rates of hyperbolic tetrahedral Coxeter groups.

Dynamical degrees of automorphisms on abelian varieties

For any given Salem number, we construct an automorphism on a simple abelian variety whose first dynamical degree is the square of the Salem number. Our construction works for both simple abelian varieties with totally indefinite quaternion multiplication and for simple abelian varieties of the second kind. We then give a complete classification of the dynamical degree sequences for abelian varieties of dimension at most four and obtain an ergodic result for sequences of pullbacks of forms.

Topological Entropy of Pseudo-Anosov Maps from a Typical Thurston’s Construction

Abstract In this paper, we develop a way to extract information about a random walk associated with a typical Thurston’s construction. We first observe that a typical Thurston’s construction entails a free group of rank 2. We also present a proof of the spectral theorem for random walks associated with Thurston’s construction that have finite 2nd moment with respect to the Teichmüller metric. Its general case was remarked by Dahmani and Horbez. Finally, under a hypothesis not involving moment conditions, we prove that random walks eventually become pseudo-Anosov. As an application, we first discuss a random analogy of Kojima and McShane’s estimation of the hyperbolic volume of a mapping torus with pseudo-Anosov monodromy. As another application, we discuss non-probabilistic estimations of stretch factors from Thurston’s construction and the powers for Salem numbers to become the stretch factors of pseudo-Anosovs from Thurston’s construction.

Counting Salem Numbers of Arithmetic Hyperbolic 3-Orbifolds

It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic 3-dimensional orbifold defines c Q^{1/2} + O(Q^{1/4}) square-rootable Salem numbers of degree 4 which are less than or equal to Q. This quantity can be compared to the total number of such Salem numbers, which is shown to be asymptotic to frac{4}{3}Q^{3/2}+O(Q). Assuming the gap conjecture of Marklof, we can extend these results to compact arithmetic 3-orbifolds. As an application, we obtain lower bounds for the strong exponential growth of mean multiplicities in the geodesic spectrum of non-compact even dimensional arithmetic orbifolds. Previously, such lower bounds had only been obtained in dimensions 2 and 3.

Mahler measures of Pisot and Salem type numbers

In this paper we consider the set of Mahler measures of generalized Pisot and Salem numbers and give a characterization of such numbers in terms of their minimal polynomials. We show that every real quadratic algebraic integer β can be expressed by the difference of the Mahler measure of a generalized Pisot number and a Pisot number in ℚ(β).

Computing Garsia entropy for Bernoulli convolutions with algebraic parameters * *Research of K G Hare was supported by NSERC Grant 2019-03930.

We introduce a parameter space containing all algebraic integers β ∈ (1, 2] that are not Pisot or Salem numbers, and a sequence of increasing piecewise continuous function on this parameter space which gives a lower bound for the Garsia entropy of the Bernoulli convolution ν β . This allows us to show that dimH(ν β ) = 1 for all β with representations in certain open regions of the parameter space.

On Salem numbers which are Mahler measures of nonreciprocal $2$-Pisot numbers

On Salem numbers which are Mahler measures of nonreciprocal $2$-Pisot numbers