Research proof index

A finite-field density-increment lemma. If the normalized additive correlation of $A$ with a set $C$ of density at least $\gamma$ differs from its random value by at least $\varepsilon$, then there is a subspace $V$ of explicitly bounded codimension. Averaging $1_A$ over $V$ raises its $L^\infty$ density to at least $(1+\varepsilon/32)\alpha$, where $\alpha$ is the density of $A$.

Differentiability of t ↦ jacobiTheta₂(t/2, t) at points in the upper half-plane.

The converse direction: DifferentiableAt on ℂ implies MDifferentiableAt on ℍ.

At every real $x$ for which $e^{ix}\ne1$, the finite-sum definition of the $N$th Dirichlet kernel agrees with the project’s closed-form expression:

$D_N(x)=D'_N(x).$

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

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

A special case of Young's convolution inequality.