Research proof index

Proposition 2.5. The bulk of the proof is in the section NiceFactorization.

Some basic consequences of Proposition 2.5, phrased in a way that make them more useful in the proof of Proposition 2.6.

If a tile $p$ lies in the $n$th minimal layer of a set of tiles $A$, then there is a tile $p'$ in the zeroth minimal layer with $p'\le p$, and the scale of $p$ is at least the scale of $p'$ plus $n$.

Andrew's Lemma : Density for algebraic extensions.

Moment generating function for count

Let $V$ be the intersection of the kernels of a set of additive characters $\Delta$. Averaging $f$ over $V$ equals the average of $\widehat f(\psi)$ over all characters, with the Fourier coefficient retained precisely when $\psi$ lies in the additive subgroup generated by $\Delta$.