Bilinear sum

A bilinear sum has the form ${\sum }_{m,n}{a}_m{b}_{n}K(m,n)$, with two independently controlled coefficient sequences. Cauchy-Schwarz, completion, and large-sieve estimates exploit this separation of variables.

A dyadic decomposition partitions a positive range into intervals $(M,2M]$. It reduces a variable-length sum to $O(\log N)$ sums in which every variable has a fixed order of magnitude.