Research proof index

A pair of measures minimizing $\tau$ exists.

For p.η ≤ 1/8, there exist τ-minimizers X₁, X₂ at zero Rusza distance. For p.η < 1/8, all minimizers are fine, by tau_strictly_decreases'. For p.η = 1/8, we use a limit of minimizers for η < 1/8, which exists by compactness.

If $d[X_1;X_2] > 0$ then there are $G$-valued random variables $X'_1, X'_2$ such that $\tau[X'_1;X'_2] < \tau[X_1;X_2]$. Phrased in the contrapositive form for convenience of proof.

For a finite group of odd order and a three-term-progression-free set $s$, the normalized inner product between $\mu_s*\mu_s$ and the uniform measure on $2s$ is exactly $|s|^{-2}$. The identity records the precise normalized count forced by the absence of nontrivial three-term progressions.

If G is torsion-free and X, Y are G-valued random variables then d[X; 2Y] ≤ 5d[X; Y].

Polynomial Freiman-Ruzsa theorem for bounded-torsion groups. Let $G$ be a finite abelian group in which $mx=0$ for every $x$, with $m\ge2$. If $A\subseteq G$ is nonempty and $|A+A|\le K|A|$, then there are a subgroup $H\le G$ and a set $c$ such that $A\subseteq c+H$, $|H|\le|A|$, and $|c|<mK^{256m^3+1}$.