Beta expansion of Salem numbers approaching Pisot numbers with the finiteness property

2015 ◽  
Vol 168 (2) ◽  
pp. 107-119
Author(s):  
Hachem Hichri
Keyword(s):  
Finiteness Property ◽  
Pisot Numbers ◽  
Salem Numbers

scholarly journals Salem Numbers and Pisot Numbers via Interlacing

2012 ◽  
Vol 64 (2) ◽  
pp. 345-367 ◽  
Author(s):  
James McKee ◽  
Chris Smyth
Keyword(s):  
Unit Circle ◽  
Rational Functions ◽  
General Construction ◽  
Pisot Numbers ◽  
Salem Numbers ◽  
Limit Points ◽  
Zeros And Poles

Abstract We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the “obvious” limit points of the set of Salem numbers produced by our theorems and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we produce all Salem numbers via an interlacing construction.

The zeta function of the beta transformation

1994 ◽  
Vol 14 (2) ◽  
pp. 237-266 ◽  
Author(s):  
Leopold Flatto ◽  
Jeffrey C. Lagarias ◽  
Bjorn Poonen
Keyword(s):  
Continuous Function ◽  
Zeta Function ◽  
Symbolic Dynamics ◽  
Counting Function ◽  
Minimum Modulus ◽  
Pisot Numbers ◽  
Salem Numbers ◽  
Integer Base ◽  
Image Position ◽  
Totally Disconnected

AbstractThe β-transformation ƒβ(x) = βx(mod 1), for β > 1, has a symbolic dynamics generalizing radix expansions to an integer base. Two important invariants of ƒβ are the (Artin-Mazur) zeta functionwhere Pk counts the number of fixed points of , and the lap-counting function where Lk counts the number of monotonic pieces of the kth iterate . For β-transformations these functions are related by ζβ(z) = (1 − z)Lβ(z). The function ζβ(z) is meromorphic in the unit disk, is holomorphic in {z: |z| < 1/β}, has a simple pole at z = 1/β, and has no other singularities with |z| = 1/β. Let M(β) denote the minimum modulus of any pole of ζβ(z) in |z| < 1 other than z = 1/β, and set M(β) = 1 if no other pole exists with |z| < 1. Then Pk = βk + O((M(β)−1+ε)k) for any ε > 0. This paper shows that M(β) is a continuous function, that ( for all β, and that An asymptotic formula is derived for M(β) as β → 1+, which implies that M(β) < 1 for all β in an interval (1, 1 + c0). The set is shown to have properties analogous to the set of Pisot numbers. It is closed, totally disconnected, has smallest element ≥ 1 + C0 and contains infinitely many β falling in each interval [n, n + 1) for n ∈ ℤ+. All known members of are algebraic integers which are either Pisot or Salem numbers.

Pisot Numbers, Salem Numbers and Distribution Modulo 1

1992 ◽  
pp. 77-99
Author(s):  
M. J. Bertin ◽  
A. Decomps-Guilloux ◽  
M. Grandet-Hugot ◽  
M. Pathiaux-Delefosse ◽  
J. P. Schreiber
Keyword(s):  
Distribution Modulo 1 ◽  
Pisot Numbers ◽  
Salem Numbers ◽  
Distribution Modulo

Pisot Numbers from {0, 1}-Polynomials

2010 ◽  
Vol 53 (1) ◽  
pp. 140-152
Author(s):  
Keshav Mukunda
Keyword(s):  
Unit Disk ◽  
Minimal Polynomial ◽  
Algebraic Integer ◽  
Pisot Number ◽  
Open Unit ◽  
Closed Unit Disk ◽  
Pisot Numbers ◽  
Salem Numbers ◽  
Salem Number ◽  
Newman 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.

Salem numbers as Mahler measures of Gaussian Pisot numbers

2020 ◽  
Vol 194 (4) ◽  
pp. 383-392
Author(s):  
Toufik Zaïmi
Keyword(s):  
Pisot Numbers ◽  
Salem Numbers

scholarly journals Salem Numbers, Pisot Numbers, Mahler Measure, and Graphs

2005 ◽  
Vol 14 (2) ◽  
pp. 211-229 ◽  
Author(s):  
James McKee ◽  
Chris Smyth
Keyword(s):  
Mahler Measure ◽  
Pisot Numbers ◽  
Salem Numbers
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