Sequences that omit a box (modulo 1)

Let be a strictly increasing sequence of real numbers satisfying(0.1)aj+1−aj?σ>0. For an open box I in 0,1d⁾, we write It is shown that the Hausdorff dimension of is d−1 whenever The case d=1 is due to Boshernitzan. The proof builds on his approach.Now let S₁,…,Sd be strictly increasing in N. Define to be the set of x in 0, 1) for which A sequence S is said to fulfill condition D(C) if it containsBr=ur,vr]∩S for which vr−ur→∞ and1+vr−ur?C#(Br). Kaufman has shown that is countable whenever S₁,…,Sd fulfill condition D(C). Here it is shown that is finite under this hypothesis. An upper bound for is provided.