We prove that BPP has Lutz's p-dimension at most 1/2 unless BPP equals EXP.
Next we show that BPP has Lutz's p-dimension zero unless BPP equals EXP
on infinitely many input lengths.
We also prove that BPP has measure zero in the smaller complexity
class SUBEXP unless MA=EXP.
Finally we show that under the plausible assumption
Dtime (2^{dn}) is hard to approximate by sat-oracle circuits
of size q (for every fixed polynomial q) bpp has p-dimension zero.