By Raphael Salem
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Additional resources for Algebraic Numbers and Fourier Analysis
Let P(x) be a polynomial in a field k. Let K be an extension of k such that, in K, P(x) can be factored into linear factors. , such that k C K' C K), the field K is said to be a splitring field of P(x), and the roots of P(x) generate K. Let a],. , a, be the roots of P(x) in the splitting field K = k(al, . , a"). , each automorphism of K whose restriction and hl, . , h, are rational integers not all 0. If wl, w2, . , up, and 1 are linearly independent, the vector (nul, . , mp) is uniformly distributed modulo 1 (see ).
What is the behavior of the product as u -,oo? Can this product tend to zero? Example, 6 = 3, (2 = 3. This may have an application to the problem of sets of multiplicity. t If r,r2+ 0, this set would be a set of multiplicity. t See Kahane and Salem, Colluquium Mathemu~icwn,Vol. V1 (1958). p. 193. By E(&) + E ( 6 ) we denote the set of all numbers XI + xrsuch that XI E E(&) and x: E E(&. + + 63 4. t Determine the values of [ for which one or the other case arises. (Of course, if f - I € S, r(u) # o(1) and the measure is purely singular.
66 Appendix 8. Kronecker's theorem. See . In the form in which we use it in Chapter 111 it may be stated as follows: are linearly independent, a l , a*, there exist integers . , k). ) uniform distribution modulo 1 of the vector (no,, .. , 9. We had occasion to cite Minkowski's theorem on linear forms in Chapters I, 111, and V1. We restate it here as follows. Ler . be n linearforms of the n variables X I , . , x, where the cwficients are real and the determinant D of the forms is not zero. There exists a point x with integral coordinates not all zero, x,, .
Algebraic Numbers and Fourier Analysis by Raphael Salem