Let $b$ be an irrational number, and $f:\mathbb R\to\mathbb R_{\geqslant0}$ be an integrable function with period $1$ such that $\displaystyle\int_0^1 f(x)\,\mathrm dx = 1$. Define$$A:=\left\{y\in[0,1]:\int_y^{y+b}f(t)\,\mathrm dt\geqslant b\right\}.$$What is the smallest possible Lebesgue measure of $A$?
When $b$ is a rational number, if we write $b=m/n$ with $\gcd(m,n)=1$, then the answer is $1/n$. The proof for the lower bound doesn't work for irrational numbers, so I would guess that the Lebesgue measure of $A$ can be arbitrarily close to $0$.