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By Raphael Salem

ISBN-10: 053498049X

ISBN-13: 9780534980498

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Let a,, a%. , an be natural integers, not all zero. We have If k + a,, the last parenthesis tends to zero. On the other hand, the first parenthesis equals and its absolute value increases infinitely with k, since, 8 being of degree n, we have This completes the proof. Remark. We have just proved that if 8 belongs to the class S and has degra n the set E([) is of the type H(*). But it does not follow that E cannot be of a simpler type. Thus, for instance, if 8 is quadratic, our theorem shows that E is of the type H").

30. , Vorlesungen Ubcr die Theorie der AIgebraischen ZahIen, 2. Auflage. Leipzig, 1954. See p. 116. ,Comptes Rendus, t Vol. 248 (1959), p. 2943. [6] KAHANE, (71 KOKSMA, J. ,Compositio mathematics, Vol. 2 (1935), p. 250. , Comptes Rendus,t Vol. 248 (1959), p. 2155. , Aufgaben und Lehrsiitze aus der Analysis, 2 vols. Berlin: Julius Springer, 1925. See Vol. 11, pp. 149-150. , Annuli dl Pisa, Vol. 7 (1938). pp. 205-248. [Ill Plsar. , Annales ScientiJques de r&ole Normale SupPrleure, Vol. 70 (1953).

Are algebraic and belong to the field of X. Some Unsolved Problems SOME UNSOLVED PROBLEMS 1. The following problem has already been quoted in Chapter I : Suppose that the real number 8 > 1 is such that there exists a real X with the property that 11 XB" )I 4 0 as the integer n increases infinitely (without any other hypothesis). Can one conclude that 8 belongs to the class S? Another way to state the same problem is: Among the numbers 8 > 1 such that, for a certain real A, 11 X8" 11 -,0 as n -, oo, do there exist numbers 8 which are not algebraic?

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Algebraic Numbers and Fourier Analysis by Raphael Salem

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