By Raphael Salem
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Extra info for Algebraic Numbers and Fourier Analysis
Then the elements of R are rational integers or integers of an imaginary quadratic field (see ). 5. There exist only a finite number of algebraic integers of given degree n, which lie with all their conjugates in a bounded domain of the complex plane (= ~91). the integral being taken in the p-dimensional unit torus P, H. Weyl's criterion becomes lim elri(HVI) + . + e2ri(HV,) n - 0 where (HV,) is the scalar product 6. 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.
Every set E of the type H(") is a set of uniqueness. PROOF. We can again suppose that E is closed, and we shall take n = 2, the two-dimensional case being typical. Suppose that the family of vectors + and = X(n& = m y,em*&. The sequence of functions (Xk(x)) satisfy the conditions (I), (2), (3), (4) of Theorem I. @) = 0 if nk,/'n, we see that the conditions (3) are satisfied, with is normal. We can assume that A consists of the points (xl, x2)such that the intervals (al, 8,) and (a9, &) being contained in (0,2 ~ ) .
We proceed now to prove that the preceding conditions are sufficient in order that E be a U-set. The points of this set E' are given by where the ej are either 0 or 1. We can, as in the case of symmetrical perfact sets, d&ne a measure carried by this set and prove that its Fourier-Stieltjes transform is fi cos 3. The homogeneous set E whose points are given by (1). where I / f = 8 is an algebraic integer of the class S and the numbers 'll, .... v d are algebraic belonging to the jeld of 8, is a set of the type H'") (n being the degree of 8), and thus a set of uniqueness.