Pisot and Salem numbers in intervals of the real line

Author:
David W. Boyd

Journal:
Math. Comp. **32** (1978), 1244-1260

MSC:
Primary 12A15

DOI:
https://doi.org/10.1090/S0025-5718-1978-0491587-8

MathSciNet review:
0491587

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Based on the work of Dufresnoy and Pisot, we develop an algorithm for determining all the Pisot numbers in an interval of the real line, provided this number is finite. We apply the algorithm to the problem of determining small Salem numbers by Salem's construction, and to the proof that certain Pisot sequences satisfy no linear recurrence relation.

**[1]**M. AMARA, "Ensembles fermés de nombres algébriques,"*Ann. Sci. École Norm. Sup.*(3), v. 83, 1966, pp. 215-270. MR**0237459 (38:5741)****[2]**P. E. BLANKSBY & H. L. MONTGOMERY, "Algebraic integers near the unit circle," Acta Arith., v. 18, 1971, pp. 355-369. MR**0296021 (45:5082)****[3]**D. W. BOYD, "Pisot sequences which satisfy no linear recurrence,"*Acta Arith.*, v. 32, 1977, pp. 89-98. MR**0427241 (55:276)****[4]**D. W. BOYD, "Small Salem numbers,"*Duke Math. J.*, v. 44, 1977, pp. 315-328. MR**0453692 (56:11952)****[5]**D. W. BOYD, "Pisot numbers and the width of meromorphic functions." (Privately circulated manuscript.)**[6]**D. G. CANTOR, "On families of Pisot*E*-sequences,"*Ann. Sci École Norm. Sup.*(4), v. 9, 1976, pp. 283-308. MR**0417115 (54:5175)****[7]**J. DUFRESNOY & Ch. PISOT, "Étude de certaines fonctions méromorphes bornées sur le cercle unité, application à un ensemble fermé d'entiers algébriques,"*Ann. Sci. École Norm. Sup.*(3), v. 72, 1955, pp. 69-92. MR**0072902 (17:349d)****[8]**J. DUFRESNOY & Ch. PISOT, "Sur les elements d'accumulation d'un ensemble fermé d'entiers algébriques,"*Bull. Sci. Math.*(2), v. 79, 1955, pp. 54-64. MR**0073634 (17:463a)****[9]**P. H. GALYEAN,*On linear recurrence relations for E-sequences*, Thesis, University of California, Los Angeles, 1971.**[10]**M. GRANDET-HUGOT, "Ensembles fermés d'entiers algébriques,"*Ann. Sci. École Norm. Sup.*(3), v. 82, 1965, pp. 1-35. MR**0193063 (33:1285)****[11]**D. E. KNUTH,*The Art of Computer Programming*. I, Addison-Wesley, Reading, Mass., 1968. MR**0378456 (51:14624)****[12]**R. SALEM, "A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan,"*Duke Math. J.*, v. 11, 1944, pp. 103-107. MR**0010149 (5:254a)****[13]**R. SALEM, "Power series with integral coefficients,"*Duke Math. J.*, v. 12, 1945, pp. 153-171. MR**0011720 (6:206b)****[14]**C. L. SIEGEL, "Algebraic integers whose conjugates lie in the unit circle,"*Duke Math. J.*, v. 11, 1944, pp. 597-602. MR**0010579 (6:39b)****[15]**C. J. SMYTH, personal communication, March 10, 1977.

Retrieve articles in *Mathematics of Computation*
with MSC:
12A15

Retrieve articles in all journals with MSC: 12A15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0491587-8

Article copyright:
© Copyright 1978
American Mathematical Society