resolution $M$ and modulus $q$ is the join of $d$ simple local factors $\mathcal{B}_i$, each of resolution $M_i$ and modulus $q_i$, where $M_i {\geqslant}M$ and $q = {\mathrm{lcm}}[q_1, \dots, q_d]$.

Local factors of large resolution and small modulus and dimension necessarily contain few sets. This fact will be useful later in the proof of Lemma \[increment lemma\].

\[atom bound\] If $\mathcal{B}$ is a local factor of dimension $d$, resolution $M$, and modulus $q$, then $$|\mathcal{B}| {\leqslant}qd{\left( \frac{N}{M} + 2 \right)}.$$

By the definition of a local factor, it suffices to bound the size of the join of $d$ simple real factors, and then bound the size of the join of $d$ simple congruence factors. The product of these two numbers gives us our final bound.

Joining $d$ congruence simple factors with moduli $q_1, \dots, q_d$ results in another congruence simple factor of modulus $q= {\mathrm{lcm}}[q_1, \dots, q_d]$. The number of parts in such a partition is $q$.

The join of $d$ simple real factors partitions $[N]$ into intervals. The upper endpoint of each of these intervals is either equal to $N$ or is equal to an endpoint of an interval in one of the original simple real factors. For a 