Simultaneous Diophantine approximation and VIP systems

Abstract

n=1 is equidistributed mod 1, which in particular implies that the density of Wα,β(f), defined to be d = limN→∞ |{x:−N≤x<N,f(x)∈(α,β)}| 2N+1 , exists and equals β − α. (Replacing the lim in the definition of density by lim sup or lim inf, one obtains the notions of upper density and lower density, respectively. Note that the family of sets having positive lower density is closed under supersets, which is a desired feature of any notion of “largeness.” Indeed, positive lower density is the first of several progressively stronger “largeness” properties that we shall be concerned with in this paper.) A set S in Z is syndetic if the union of finitely many of its additive shifts is all of Z. Alternatively, S is syndetic if it intersects non-trivially any large enough d-dimensional cube; namely, if there exists k such that for all choices of M1, . . . ,Md, S ∩ ∏d i=1[Mi,Mi + k] 6= ∅. In Z, then, S is syndetic if it intersects non-trivially any large enough interval, i.e. has bounded gaps. Syndeticity is a property that is strictly stronger than that of positive lower density and is the second notion of largeness of interest to us. Van der Corput provided the following impressive generalization of Theorem W in [VdC].