Betting strategies with bounded splits

We show that a pair of Kolmogorov-Loveland betting strategies cannot win on every non-Martin-Löf random sequence if either of the two following conditions is true: (I) There is an unbounded computable function $g$ such that both betting strategies, when betting on an infinite binary sequence, almost surely, for almost all $\mathrm{\ell }$, bet on at most $\mathrm{\ell }-g(\mathrm{\ell })$ positions among the first $\mathrm{\ell }$ positions of the sequence. (II) There is a sublinear function $g$ such that both betting strategies, when betting on an infinite binary sequence, almost surely, for almost all $\mathrm{\ell }$, bet on at least $\mathrm{\ell }-g(\mathrm{\ell })$ positions among the first $\mathrm{\ell }$ positions of the sequence.