Defining a function $f_{a,T}:\mathbb R \to \{0,1\}$ to be $T$-periodic ($\forall x: f_{a,T}(x)=f_{a,T}(x+T)$), with $a\in[0,T]$ such that $\forall x\in [0,T] : f_{a,T}(x)= 1 \iff x\in [0,a]$.
Given parameters $k,a,b, T, S$, what's the probability (over random $r$ which is uniformly distributed over $[0,T]$) that there exists an interval of size $\geq k$ (size of an interval is its endpoints difference) which is a sub-interval of the interval $[0,T]$ and on which it holds that both $f_{a,T}(x+r)=1$ and $f_{b,S}(x)=1$?
