I don't know if the following qualifies as an "interesting result" in "digit theory", but Carlo Sanna, a student from the Università di Torino, has recently published a paper in elementary number theory [1] which is concerned with properties relating arithmetic progressions and the sum-of-digit function (in an arbitrary base $b$): It includes a few references and a number of questions that you may find at least intruiging.
[1] Carlo Sanna, On Arithmetic Progressions of Integers with a Distinct Sum of Digits, Vol. 15 (2012), Article 12.8.1 (see here).