 Small Type I and Type II sums

Extend $f:[1,N]\cap \mathbb{Z}\to \mathbb{C}$ by zero, and assume $|f|\le 1$. In one dyadic endpoint convention, Type I sums are $\delta$-small if ${\sum }_{M\le m<2M}|{\sum }_{n\in I_m}f(mn)|\le \delta N$ for every dyadic $M\le N^{1/100}$ and every choice of integer intervals $I_m\subset [K,2K)$, $MK\le N$. Type II sums are $\delta$-small if $|{\sum }_{m\in [M,2M)}{\sum }_{n\in [K,2K)}{a}_m{b}_{n}f(mn)|\le \delta N$ for all dyadic $M,K\in [N^{1/100},N^{99/100}]$ and coefficients $|a_m|,|b_n|\le 1$. Empty tails are handled by zero extension.