A generalization of the 3d distance theorem

Abstract

Let P be a positive rational number. A function f:R→R has the finite gaps property mod P if the following holds: for any positive irrational α and positive integer M, when the values of f(mα), 1≤m≤M, are inserted mod P into the interval [0, P) and arranged in increasing order, the number of distinct gaps between successive terms is bounded by a constant kf which depends only on f. In this note, we prove a generalization of the 3d distance theorem of Chung and Graham. As a consequence, we show that a piecewise linear map with rational slopes and having only finitely many non-differentiable points has the finite gaps property mod P. We also show that if f is the distance to the nearest integer function, then it has the finite gaps property mod 1 with kf≤6.