Denominator-preserving maps

Original · EN

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.