Exponential decay of threshold correlations for symmetric periodic functions

We study how often $n$ distinct integer dilates of a periodic function can all lie above a fixed threshold or all lie below its negative. We consider continuous functions $f:\mathbb R\to[-1,1]$ that are $1$-periodic, even, strictly decreasing on $[0,1/2]$, and satisfy $f(0)=1$ and $f(t+1/2)=-f(t)$.