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MULTIDIMENSIONAL VAN DER CORPUT SETS AND SMALL FRACTIONAL PARTS OF POLYNOMIALS

MULTIDIMENSIONAL VAN DER CORPUT SETS AND SMALL FRACTIONAL PARTS OF POLYNOMIALS

We establish Diophantine inequalities for the fractional parts of generalized polynomials, in particular for sequences with a non-integral real number and , as well as for where runs through all prime numbers. This is related to classical work of Heilbronn and to recent results of Bergelson et al.