Georgia Institute of Technology College of Sciences

The `middle third integer Cantor set' consists of those integers which do not have a 2 in their base 3 representation. We will review and extend some results about such sets. For a general integer Cantor set K, with 0 as an allowed digit, it is known that K is intersective, a result of Furstenberg-Katznelson. That is, for a dense set of integers A,  A-A must intersect K.   Writing K={k_1, k_2, ...},  we show that the set of n such that k_n\in A-A has positive density.   The set  p(K), where p is an integer polynomial with zero constant term, is also intersective due to Bergelson-McCutcheon. We show the same density result for p(K).  We also show an L^2 Ergodic Theorem along K.  The pointwise Ergodic Theorem lies beyond current techniques.  Joint work with A Burgin, A Fragkos, D. Mena, M Reguera.