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X, such that provided that 61 - . 6, 2 1 D 1. The result holds if the coefficients aqp arc complex, provided that complex forms figure in conjugate pairs, and that the two 6,'s corresponding to a conjugate pair are equal. The theorem is usually proved by using the following result. If K is a convex region of volume V in the Euclidean space R* with center of symmetry at the origin and if V > 2*, the region K contains points of integral coordinates other than the origin. An extremely elegant proof of this result has been given by C.

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). pp. 105-133. ,Duke Mathematical Journal, VOI. 11 (1944), pp. 103-108. ,Duke Mathsmaticaf Journal, Vol. 12 (1945). pp. 153-172.

We shall denote by X(x) and y(x) respectively two functions constructed with respect to the intervals (a,, bl) and ( a , 62) as was, in the case of sets H, the function X(x) with respect to (a, 6). Under these conditions, the functions are equal to zero for all k and all x E E. This sequence of functions will play the role of the sequence denoted by L(x) in Theorem I. Thus. the condition (I) of that theorem is satisfied. Write X(X) = which is positive if e has been chosen small enough. Second application.

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