Denominator-preserving maps

Abstract

Let F be a continuous injective map from an open subset of Rn to Rn. Assume that, for infinitely many k>1, F induces a bijection between the rational points of denominator k in the domain and those in the image (the denominator of (a1/b1,...,an/bn) being the l.c.m. of b1,...,bn). Then F preserves the Lebesgue measure.

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