Research proof index

For every $\delta,\varepsilon>0$, there is an explicit positive uniform bound $C(\delta,\varepsilon)$ such that, if a measurable $2\pi$-periodic function $g$ satisfies $\lVert g(x)\rVert\le C(\delta,\varepsilon)$ for every $x$, then the set where $\sup_N\lVert S_Ng(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$.