i) 1. Introduction.Let C be the additive group of real numbers modulo 1, and let x -* {x} natural mapping from reals onto C. It is clear what we shall mean by an interval / in length 1(1) /.Denote distance number a to closest integer || ||.The image set Ç satisfying £ -01| ^ s with given 0 < 1/2 example

We shall prove theorems on simultaneous approximation which generalize Roth's well-known theorem [3] rational to a single algebraic irrational ~.Throughout the paper, []~]1 will denote distance from real number ~ nearest integer.TH~.OREM 1.Let ax ..... an be numbers such that 1, al 0~ are linearly independent over field Q o/rationals.Then/or every e > 0 there only finitely many positive integers q with COROLLARY.Suppose o~ 1 .... , o:~, as above.There n-tuples (Pl/q p~/q) o/rationals...

Journal Article IRREGULARITIES OF DISTRIBUTION Get access WOLFGANG M. SCHMIDT University of ColoradoBoulder, Colorado Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Mathematics, Volume 19, Issue 1, 1968, Pages 181–191, https://doi.org/10.1093/qmath/19.1.181 Published: 01 January 1968 history Received: 16 June 1967

The basic problem of diophantine approximations could be formulated as follows. Assume a Cartesian coordinate system in euclidean E2 to given. Let L line through the origin 0. Find L' 0 and defined over rationals Q (defined by linear equation with rational coefficients) which is close possible L. As measure their closeness, introduce /r(L, L') = sin Ap, where p angle between L', 0, there will infinitely many such lines satisfying J(L, not rationals, are !(L, < (5-1/2 + 8) H(L)-2 This...

Introduction.A basic result in the Geometry of Numbers is Minkowski's Second Convex Body Theorem.Given a closed symmetric convex body K R n and lattice Λ , ith successive minimum λ i 1 ≤ n, with respect to least number > 0 for which λK contains linearly independent points.Clearly,where Vol(K) volume det determinant Λ. Suppose µ . ., are reals + • = 0, Q let T : → be linear map withThen gives rise bodies K(Q) := (K) parametrized by Q.We propose study minima (Q), (Q) K(Q), as functions Q....

for the number of lattice-points in S. Here, and throughout this paper, a lattice-point is point with integral coordinates. If S Borel set finite volume V(S), one would expect that L(S) about same order magnitude as V(S). Hence we define discrepancy D(S) by (1) = I L(S)V(S)1 As companion L(S), introduce P(S), primitive (A primitive, if its coordinates are relatively prime.) We put (2) E(S) P(S)t(n)V(S)'. I| Next, let 1' be family sets volumes, such (i) SeC, TCb, then either SOT or TCS. (ii)...

Let C be a compact convex body in R⊃m and consider set of points selected at random from according to some well–behaved sampling distribution. We obtain an asymptotic expression for the positive moments kth near–neighbour distance distribution as number increases infinity.