Research proof index

Lemma 3.3 (Part 2): Bound on agreeing cells inside D For any column j that is in the disagreement set D, there is at most one parameter r in R* such that the columns Uᵣ j and Vᵣ j agree. Therefore, the total number of agreeing cells (r, j) with j ∈ D is at most |D|.

Lemma 3.3 (Part 1): Bound on agreeing cells outside D The set of agreeing cells (r, j) where j ∉ D is exactly the Cartesian product of R* and Dᶜ (the columns not in D).

The number of elements in ∀ i, ↥(S i) satisfying eval (↑·) f = 0 equals the number of elements in Fintype.piFinset (fun i => (S i).toFinset) satisfying eval · f = 0.

In the ambient finite $\mathbb F_2$-vector space, exactly half of the additive homomorphisms $\varphi:G\to\mathbb F_2$ take a fixed nonzero vector $x$ to $1$: $2\,|\{\varphi:\varphi(x)=1\}|=|G|$.

For every set $A$ in the ambient finite $\mathbb F_2$-vector space, some linear functional $\varphi:G\to\mathbb F_2$ has at least $(|A|-1)/2$ elements of $A$ in its $1$-fiber.

The real-line Carleson operator is positively homogeneous. For every $a>0$,

$T f(x)=a\,T(f/a)(x),$

where the scalar on the right is interpreted in the extended nonnegative reals.