Seventy years of Salem numbers

I survey results about, and recent applications of, Salem numbers.

[1]  Yves Meyer,et al.  Algebraic numbers and harmonic analysis , 1972 .

[2]  B. Solomyak,et al.  On the modulus of continuity for spectral measures in substitution dynamics , 2013, 1305.7373.

[3]  Artūras Dubickas Sumsets of Pisot and Salem numbers , 2008 .

[4]  Paul Reschke Salem Numbers and Automorphisms of Complex Surfaces , 2012, 1202.5245.

[5]  A. Kolpakov,et al.  The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space , 2014, Canadian Journal of Mathematics.

[6]  Qiang Wu,et al.  THE TRACE PROBLEM FOR TOTALLY POSITIVE ALGEBRAIC INTEGERS , 2011, Journal of the Australian Mathematical Society.

[7]  M. Gromov,et al.  L'enseignement Mathématique on the Entropy of Holomorphic Maps , 2022 .

[8]  David W. Boyd,et al.  Speculations Concerning the Range of Mahler's Measure , 1980, Canadian Mathematical Bulletin.

[9]  M. E. Bertin,et al.  MAHLER MEASURE OF MULTIVARIABLE POLYNOMIALS , 2012 .

[10]  Curtis T. McMullen,et al.  Automorphisms of even unimodular lattices and unramified Salem numbers , 2002 .

[11]  Salem numbers defined by Coxeter transformation , 2010 .

[12]  Michael J. Mossinghoff,et al.  Negative Pisot and Salem numbers as roots of Newman polynomials , 2014 .

[13]  C. Doche,et al.  EQUIDISTRIBUTION MODULO 1 AND SALEM NUMBERS , 2008 .

[14]  James McKee,et al.  Salem Numbers of Trace -2 and Traces of Totally Positive Algebraic Integers , 2004, ANTS.

[15]  Sur la distribution de certaines séries aléatoires , 1971 .

[16]  Yuriko Umemoto The growth function of Coxeter dominoes and 2-Salem numbers , 2014 .

[17]  Steve Fisk A very short proof of Cauchy's interlace theorem for eigenvalues of Hermitian matrices , 2005 .

[18]  Ruth Kellerhals,et al.  On the growth of cocompact hyperbolic Coxeter groups , 2009, Eur. J. Comb..

[19]  A. Reid,et al.  The Arithmetic of Hyperbolic 3-Manifolds , 2002 .

[20]  F. Amoroso UNE MINORATION POUR L'EXPOSANT DU GROUPE DES CLASSES D'UN CORPS ENGENDRÉ PAR UN NOMBRE DE SALEM , 2007 .

[21]  A. Dubickas Arithmetical Properties of Powers of Algebraic Numbers , 2006 .

[22]  M. Bertin,et al.  Conjecture de Lehmer et petits nombres de Salem , 1989 .

[23]  Susan G. Williams,et al.  Mahler Measure of Alexander Polynomials , 2001, math/0105234.

[24]  C. McMullen K3 surfaces, entropy and glue , 2011 .

[25]  Hachem Hichri Beta expansion for some particular sequences of Salem numbers , 2014 .

[26]  Michael J. Mossinghoff Polynomials with small Mahler measure , 1998, Math. Comput..

[27]  Chris Smyth,et al.  On the lines passing through two conjugates of a Salem number , 2008, Mathematical Proceedings of the Cambridge Philosophical Society.

[28]  R. Salem,et al.  A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan , 1944 .

[29]  Arithmetic groups and Salem numbers , 1992 .

[30]  Algebraic integers with two conjugates outside the unit circle , 1953 .

[31]  Curtis T. McMullen,et al.  Dynamics on K3 Surfaces: Salem Numbers and Siegel Disks , 2002 .

[32]  Nick Lord,et al.  Pisot and Salem Numbers , 1991 .

[33]  Y. Peres,et al.  Sixty Years of Bernoulli Convolutions , 2000 .

[34]  Curtis T. McMullen,et al.  Coxeter groups, Salem numbers and the Hilbert metric , 2002 .

[35]  On groups generated by two positive multi-twists: Teichmuller curves and Lehmer's number , 2003, math/0304163.

[36]  Toufik Zaïmi An arithmetical property of powers of Salem numbers , 2006 .

[37]  Chris Smyth Salem numbers of negative trace , 2000, Math. Comput..

[38]  David W. Boyd,et al.  A Characterization of Two Related Classes of Salem Numbers , 1995 .

[39]  Thomas Zehrt,et al.  The growth function of Coxeter garlands in $${\mathbb{H}^{4}}$$ , 2012 .

[40]  D. Cantor On power series with only finitely many coefficients(mod 1): Solution of a problem of Pisot and Salem , 1977 .

[41]  K. Schmidt,et al.  On Periodic Expansions of Pisot Numbers and Salem Numbers , 1980 .

[42]  David W. Boyd,et al.  On the beta expansion for Salem numbers of degree 6 , 1996, Math. Comput..

[43]  James McKee,et al.  Integer symmetric matrices having all their eigenvalues in the interval [−2,2] , 2007, 0705.3599.

[44]  Répartition (mod 1) des puissances successives des nombres réels , 1946 .

[45]  Changan SOME PROPERTIES OF THE SPECTRUM OF GRAPHS , 1999 .

[46]  Eriko Hironaka,et al.  The arithmetic and geometry of Salem numbers , 2001 .

[47]  S. Akiyama,et al.  Salem numbers and uniform distribution modulo 1 , 2005 .

[48]  E. Hironaka The Lehmer Polynomial and Pretzel Links , 2001, Canadian Mathematical Bulletin.

[49]  Eric Bedford,et al.  Periodicities in linear fractional recurrences: Degree growth of birational surface maps , 2005 .

[50]  Mohamed Kerada Une caractérisation de certaines classes d'entiers algébriques généralisant les nombres de Salem , 1995 .

[51]  W. Parry On theβ-expansions of real numbers , 1960 .

[52]  There are Salem Numbers of Every Trace , 2003, math/0308213.

[53]  Klaus Schmidt,et al.  Algebraic Coding of Expansive Group Automorphisms and Two-sided Beta-Shifts , 2000 .

[54]  T. Vijayaraghavan On the fractional parts of the powers of a number (II) , 1940, Mathematical Proceedings of the Cambridge Philosophical Society.

[55]  J. McKee,et al.  Galois theory of Salem polynomials , 2009, Mathematical Proceedings of the Cambridge Philosophical Society.

[56]  R. Kellerhals Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers , 2013 .

[57]  Corps de nombres engendrés par un nombre de Salem , 1999 .

[58]  A. Garsia Arithmetic properties of Bernoulli convolutions , 1962 .

[59]  David W. Boyd Pisot and Salem numbers in intervals of the real line , 1978 .

[60]  C. McMullen Automorphisms of projective K3 surfaces with minimum entropy , 2016 .

[61]  A Salem number with degree 34 and trace −3 , 2015 .

[62]  L. Kronecker,et al.  Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten. , 1857 .

[63]  Integer symmetric matrices of small spectral radius and small Mahler measure , 2009, 0907.0371.

[64]  R. Salem,et al.  Power series with integral coefficients , 1945 .

[65]  On the growth of hyperbolic 3-dimensional generalized simplex reflection groups , 2012 .

[66]  R. Salem Sets of uniqueness and sets of multiplicity , 1943 .

[67]  Salem numbers from a class of star-like trees , 2015, 1501.05557.

[70]  K. Schmidt,et al.  Symbolic representations of nonexpansive group automorphisms , 2004, math/0409257.

[71]  Kevin G. Hare BETA-EXPANSIONS OF PISOT AND SALEM NUMBERS , 2007 .

[72]  Thomas Zehrt,et al.  The growth function of Coxeter garlands in IH^4 , 2012 .

[73]  D. Boyd TRANSCENDENTAL NUMBERS WITH BADLY DISTRIBUTED POWERS , 1969 .

[74]  Y. Bugeaud Distribution Modulo One and Diophantine Approximation: References , 2012 .

[75]  Walter D. Neumann,et al.  Arithmetic of Hyperbolic Manifolds , 1992 .

[76]  J. McKee,et al.  Salem Numbers and Pisot Numbers via Interlacing , 2010, Canadian Journal of Mathematics.

[77]  Laurent Barthold,et al.  Salem Numbers and Growth Series of Some Hyperbolic Graphs , 2002 .

[78]  Alexander Kolpakov,et al.  Deformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers , 2011, Eur. J. Comb..

[79]  Qiang Wu,et al.  Minimal Mahler Measures , 2008, Exp. Math..

[80]  Valérie Flammang,et al.  Trace of totally positive algebraic integers and integer transfinite diameter , 2008, Math. Comput..

[81]  C. McMullen,et al.  Cyclotomic factors of Coxeter polynomials , 2009 .

[82]  P. Erdös On a Family of Symmetric Bernoulli Convolutions , 1939 .

[83]  De-Jun Feng Multifractal analysis of Bernoulli convolutions associated with Salem numbers , 2011, 1111.2414.

[84]  David W. Boyd Small Salem numbers , 1977 .

[85]  B. Solomyak On the random series $\sum \pm \lambda^n$ (an Erdös problem) , 1995 .

[86]  Henry B. Mann,et al.  On linear relations between roots of unity , 1965 .

[87]  James McKee,et al.  Salem Numbers, Pisot Numbers, Mahler Measure, and Graphs , 2005, Exp. Math..

[88]  Curtis T. McMullen,et al.  Dynamics on blowups of the projective plane , 2007 .

[89]  S. Akiyama,et al.  Constructions of Pisot and Salem numbers with flat palindromes , 2008 .

[90]  Annette Decomps-Guilloux,et al.  Nouvelles caractérisations des nombres de Pisot et de Salem , 1985 .

[91]  K. Oguiso The third smallest Salem number in automorphisms of K3 surfaces , 2009, 0905.2396.

[92]  R. Salem Algebraic numbers and Fourier analysis , 1963 .

[93]  D. Boyd Pisot sequences which satisfy no linear recurrence II , 1977 .

[94]  Peter B. Borwein,et al.  Some computations on the spectra of Pisot and Salem numbers , 2002, Math. Comput..

[95]  F. Beukers,et al.  Cyclotomic points on curves , 2001 .