Consider a convex polytope with lattice vertices and at least one interior lattice point. We prove that the number of boundary lattice points is bounded above by a function of the dimension and the number of interior lattice points. This extends to arbitrary dimension a result of Scott for the two dimensional case.

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For any finite set A of positive integers, let E A :=[: # (0, 1): : is irrational, and every partial quotient in the (infinite) simple continued fraction expansion of : is an element of A]. For sets A with fewer than two elements, E A is uninteresting. For |A| 2, E A is a kind of Cantor fractal dust, with a Hausdorff dimension (dim E A ) between 0 and 1. This work presents an algorithm which, given a finite set A of between 2 and N positive integers 2 N , determines dim E A to within \2 &N using O(N 7 ) elementary bit operations. There is also a convenient implementation of the algorithm in Mathematica code, together with a small table and some conjectures.