Research proof index

There is a measurable exceptional set $G'\subseteq G$ with $2\mu(G')\le\mu(G)$ such that every measurable $f$ bounded by $\mathbf{1}_F$ satisfies

$\int_{G\setminus G'}^+\lVert\operatorname{CarlesonSum}f(x)\rVert\,dx\le C(a,q)\mu(G)^{1-1/q}\mu(F)^{1/q}.$

Lemma: Distance of Affine Combination is Bounded by Interleaved Distance

Lemma 4.3, [AHIV22] (row-span lower bound). If the interleaved word U⋆ is more than e far from the interleaved code L^⋈κ, then the row-span of U⋆ contains a word more than e far from L. The additional field-size assumption |F| > e is needed: for small fields one can have a linear subspace of F^ι consisting entirely of e-sparse vector...

If $X_1, X_2$ is a $\phi$-minimizer, then $d[X_1;X_2] = 0$.

Let $T_1,T_2,T_3$ be measurable random variables in a finite abelian group with $T_1+T_2+T_3=0$, and set $\delta=I[T_1:T_2]+I[T_1:T_3]+I[T_2:T_3]$. For any measurable $Y_1,\ldots,Y_n$ and any $\alpha>0$, there is a measurable random variable $U$ such that $d[U;U]+\alpha\sum_{i=1}^n d[Y_i;U]\le\left(2+\frac{\alpha n}{2}\right)\delta+\alpha\sum_{i=1}^n d[Y_i;T_2].$

Helper Lemma relating row distance to interleaved distance (as derived from DG25): d((Uᵣ)ᵢ, C) ≤ d^m(Uᵣ, C^m)