The arithmetic and geometry of Salem numbers

Ghate, Eknath; Hironaka, Eriko

doi:10.1090/s0273-0979-01-00902-8

Cited by 35 publications

(40 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This problem has received a considerable attention in Mahler's measure setting (see [9], [39,40], [14], [17]), and it remains a very active area of research. In particular, it is of importance to characterize multivariate polynomials with integer coefficients satisfying P n 0 = 1.…”

Section: Upper Estimate Turns Into Equality For Any Polynomial Not Vamentioning

confidence: 99%

Polynomial inequalities, Mahler's measure, and multipliers

Pritsker¹

2008

Number Theory and Polynomials

View full text Add to dashboard Cite

Abstract. We survey polynomial inequalities obtained via coefficient multipliers, for norms defined by the contour or the area integrals over the unit disk. Special attention is devoted to the Szegő composition and the inequalities related to Mahler's measure.We also consider a new height on polynomial spaces defined by the integral over the normalized area measure on the unit disk. This natural analog of Mahler's measure inherits many nice properties such as the multiplicative one. However, this height is a lower bound for Mahler's measure, and it fails an analog of Lehmer's conjecture. The Szegő composition and polynomial inequalitiesThis paper is a survey of results on polynomial inequalities obtained via coefficient multipliers, and other topics related to Mahler's measure. Let C n [z] and Z n [z] be the sets of all polynomials of degree at most n with complex and integer coefficients respectively. Mahler's measure of a polynomial P n ∈ C n [z] is defined byIt is also known as the contour geometric mean or as the H 0 Hardy space norm. The latter name is explained by the following relation to the Hardy spaces. Defining the Hardy space norm bywe note [18] that M (P n ) = lim p→0+ P n H p . An application of Jensen's inequality immediately gives that

show abstract

Section: Upper Estimate Turns Into Equality For Any Polynomial Not Vamentioning

confidence: 99%

Polynomial inequalities, Mahler's measure, and multipliers

Pritsker¹

2008

Number Theory and Polynomials

View full text Add to dashboard Cite

show abstract

“…Since there are only a finite number of monic polynomials of bounded degree with bounded Mahler measure [4], the set of complex Salem numbers of bounded degree over Q has the least element. Consequently, we have Corollary 3.4.…”

Section: Uniform Lower Bound For Systoles Of Non-uniform Arithmetic Mmentioning

confidence: 99%

Systole on locally symmetric spaces

Ким

2020

Bull. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Investigations into different subclasses of the Salem numbers can be found in [21,12], and a survey of some recent appearances of Salem numbers in geometry and arithmetic can be found in [30].…”

Section: Definition 23mentioning

confidence: 99%

“…The least Pisot number is the so- is not even known whether the class of Salem numbers is bounded properly away from 1 [30].…”

Section: Definition 23mentioning

confidence: 99%

An experimental investigation of the normality of irrational algebraic numbers

Nielsen

Simonsen

2013

Math. Comp.

View full text Add to dashboard Cite

Abstract. We investigate the distribution of digits of large prefixes of the expansion of irrational algebraic numbers to different bases.We compute 2·3 18 bits of the binary expansions (corresponding to 2.33·10 8 decimals) of the 39 least Pisot-Vijayaraghavan numbers, the 47 least known Salem numbers, the least 20 square roots of positive integers that are not perfect squares, and 15 randomly generated algebraic irrationals. We employ these to compute the generalized serial statistics (roughly, the variant of the χ 2 -statistic apt for distribution of sequences of characters) of the distributions of digit blocks for each number to bases 2, 3, 5, 7 and 10, as well as the maximum relative frequency deviation from perfect equidistribution. We use the two statistics to perform tests at significance level α = 0.05, respectively, maximum deviation threshold α = 0.05.Our results suggest that if Borel's conjecture-that all irrational algebraic numbers are normal-is true, then it may have an empirical base: The distribution of digits in algebraic numbers appears close to equidistribution for large prefixes of their expansion. Of the 121 algebraic numbers studied, all numbers passed the maximum relative frequency deviation test in all considered bases for digit block sizes 1, 2, 3, and 4; furthermore, 92 numbers passed all tests up to block size 4 in all bases considered.

show abstract

The arithmetic and geometry of Salem numbers

Cited by 35 publications

References 38 publications

Polynomial inequalities, Mahler's measure, and multipliers

Polynomial inequalities, Mahler's measure, and multipliers

Systole on locally symmetric spaces

An experimental investigation of the normality of irrational algebraic numbers

Contact Info

Product

Resources

About