= Small Type I and Type II sums
Extend $f:[1,N]\cap\mathbb Z\to\mathbb C$ by zero, and assume $|f|\le1$. 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 $|\sum_{m\in[M,2M)}\sum_{n\in[K,2K)}a_mb_nf(mn)|\le\delta N$ for all dyadic $M,K\in[N^{1/100},N^{99/100}]$ and coefficients $|a_m|,|b_n|\le1$. Empty tails are handled by zero extension.
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