A metrical theorem in geometry of numbers

Abstract

for the number of lattice-points in S. Here, and throughout this paper, a lattice-point is a point with integral coordinates. If S is a Borel set of finite volume V(S), one would expect that L(S) is of about the same order of magnitude as V(S). Hence we define the discrepancy D(S) by (1) D(S) = I L(S)V(S)1 As a companion for L(S), we introduce P(S), the number of primitive lattice-points in S. (A lattice-point is primitive, if its coordinates are relatively prime.) We put (2) E(S) = I P(S)t(n)V(S)'. I| Next, let 1' be a family of Borel sets with finite volumes, such that (i) If SeC, TCb, then either SOT or TCS. (ii) There exist SeI4 with arbitrarily large V(S). Finally, throughout this paper, +1(s), s _ 0, should be a positive, nondecreasing function, such that fo4(s)-1ds exists. THEOREM 1. Suppose n >2. Then for almost linear transformation A (almost every in the sense of the n2-dimensional euclidean metric induced by matrix-representation for A), (3) D(AS) = O(V-12J2 log V ytI2(log V)), (4) E(AS) = O(V-1/2 log V 4,1/2(log V)) for SC?. More explicitly, for almost A there exist constants ci(A), c2(A), such that D(AS) c2(A) and SE+q. In R2 our results are a little weaker: Received by the editors March 13, 1959. 516

Locations

• Transactions of the American Mathematical Society - View - PDF