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. 

In 1992, Olson and Zalik conjectured that no system of translates can be a Schauder basis for L^2(R). This conjecture remains open as of the time of writing. Although some partial results regarding Olson-Zalik conjecture have been proved to be true, a characterization of subspaces of L^2(R) that do not admit a Schauder basis, or an unconditional basis is still unknown. 

In this talk, we will begin with a brief introduction to Olson-Zalik conjecture including its recent development. Then we will show that a family of modulation spaces do not admit unconditional bases formed by a system of translates. This observation led us to the following generalized Olson-Zalik conjecture ``Assume X is a separable Banach space that is continuously embedded into L^2(R). Then X does not admit a Schauder basis of translates if it is closed under Fourier transform". Finally, we close this talk by showing that if a closed subspace of L^2(R) is closed under Fourier transform, then it does not admit a Schauder basis of certain translates.