Abstract
The statement for r = 1 is proved, with more precision, by Amice [1] and Kahane [2]. It will be clear from the proof that the inequalities in the definition of A(x) can be strengthened almost arbitrarily. First we express the exceptional set in Theorem 1 as a denumerable union of closed sets. Let U1, * * *, Uj, * * * be a sequence of open sets in Rr forming a base for the topology, and A the subgroup of RT of integral vectors. Put, for each j,
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