C. McMullen math.harvard.edu/~ctm/papers/home/text/papers/winning/… Many of the papers of David Färm arxiv.org/find/math/1/au:+Farm_D/0/1/0/all/0/1 Paper with Jimmy Tseng arxiv.org/abs/1202.4109 The following paper of Moshchevitin captures this well N. G. Moshchevitin. On sublacunary sequences and winning sets (english). Math. Notes, (3–4):592–596, 2005. An application of the previous papers to Cantor series arxiv.org/abs/1302.7064 And some of the papers of Jimmy Tseng are on this topic arxiv.org/find/math/1/au:+Tseng_J/0/1/0/all/0/1

I can give a partial list, but I want to state upfront that I'm not an expert in the area and this list will be highly skewed in favor of my own papers and results relating to normal numbers. I will say that Schmidt's result is a case of something like if you have a transformation $T$ on some space $X$, then the set of all numbers $x \in X$ where the forward orbits of $x$ under $T$ is winning.