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By Raphael Salem

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N = 1) can be written as A' (}N' I where ~ is a nonvanishing determinant depending only on () (and independent of N), say, ~ = ~((}). Minkowski's theorem can be applied, provided ~ fr nn2N> (}N' and, after choosing (J', we can always determine N so that this condition be fulfilled, since ()/2 > 1. (}mx -- P I < - 2 N1n (mod 1), that is to say (6) IX(}mx - X((} - 1) (em+l + ... 2N1n (mod 1). •, gk+n. The number of points Ok depends evidently on k, n, and the choice of the e's; but we shall prove that there are at most 2N+-n-l distinct points Ok.

A necessary and sufficient conditionfor E(~) to be a set of uniqueness is that I/~ be a number of the class S [14]. THEOREM. The necessity of the condition follows from what has been said in the preceding chapter. We have only to prove here the sufficiency: If ~-I belongs to the class S, E(~) is a U-set. We simplify the formulas a little by constructing the set E(~) on [0, 1]. We write 0 = I/~ and suppose, naturally, that 0 > 2. We assume that 0 is an algebraic integer of the class S and denote by n its degree.

Another Class of Algebraic Integers 35 is not linear. ' e'ril'~ ,.. = Jo( 47rh) is not zero for all integers h ~ o. In the general case (T not quadratic) if 2k is the degree of T, we have, using the preceding notations, I T m+ --; + T k-l k-1 L: ar + i=l L: arm = 0 i=l (mod I) or T m I + ----; + i=1 L: 2 cos 27rmWi =0 T k-1 (mod I) and we have to prove that the sequence vm = 2 cos 27rmWl + ... + 2 cos 27rmWk_l is not uniformly distributed modu10 I. We use here a lemma analogous to the preceding one.

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Algebraic numbers and Fourier analysis (Heath mathematical monographs) by Raphael Salem

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