Research proof index

A Ruzsa-distance lower bound for sums of independent copies. Let $X_1',X_2'$ be independent copies of the $\tau$-minimizers $X_1,X_2$, and set $k=d[X_1;X_2]$. Then $d[X_1+X_1';X_2+X_2']\ge k-\frac{\eta}{2}\bigl(d[X_1;X_1]+d[X_2;X_2]\bigr).$

The Marcinkiewicz-Zygmund inequality for real-valued functions, with a slightly easier to bound constant than Real.marcinkiewicz_zygmund'. Note that RCLike.marcinkiewicz_zygmund is another version that works for both ℝ and ℂ at the expense of a slightly worse constant.

The Marcinkiewicz-Zygmund inequality for real-valued functions, with a slightly better constant than Real.marcinkiewicz_zygmund.

Fix a nonempty finite set $A$ in an elementary abelian $2$-group. For any measurable random variables $Y_1,Y_2$, there are a subspace $H$ and a random variable $U$ uniformly distributed on $H$ such that the source's $\rho[\,\cdot\,\#A]$ functional satisfies $2\rho[U\#A]\le\rho[Y_1\#A]+\rho[Y_2\#A]+8d[Y_1;Y_2]$.

Rudin's inequality, exponential form.

Rudin's inequality, usual form.