Research proof index

For an antichain $\mathfrak{A}$ of pairwise incomparable tiles, and measurable functions $f$ and $g$ bounded by the indicators of $F$ and $G$, the pairing of $g$ with the Carleson sum over $\mathfrak{A}$ is controlled by the $L^2$ norms of $f$ and $g$ and by positive powers of the two tile-density parameters. Concretely, the bound is

$C(a,q)\,\mathrm{dens}_1(\mathfrak{A})^{(q-1)/(8a^4)}\,\mathrm{dens}_2(\mathfrak{A})^{1/q-1/2}\,\lVert f\rVert_2\lVert g\rVert_2.$

For an antichain $\mathfrak A$, a measurable set $A\subseteq G$, and measurable $f$ bounded by $\mathbf 1_F$, the norm of the Carleson sum has the integral estimate

$\int_A^+\lVert\operatorname{CarlesonSum}_{\mathfrak A}f(x)\rVert\,dx\le C(a,q)\,\mathrm{dens}_1(\mathfrak A)^{(q-1)/(8a^4)}\,\mathrm{dens}_2(\mathfrak A)^{1/q-1/2}\,\lVert f\rVert_2\,\mu(G)^{1/2}.$

Fix a frequency parameter $\vartheta$, a level $N$, and a spatial grid cube $L$. Among the auxiliary tiles attached to an antichain $\mathfrak{A}$ whose spatial cube is exactly $L$, the total measure of their active sets inside $G$ is at most

$2^{a(N+5)}\,\mathrm{dens}_1(\mathfrak{A})\,\mu(L).$

If $F$ is a modular form where $F(it)$ is positive for sufficiently large $t$ (i.e. constant term is positive) and the derivative is positive, then $F$ is also positive.

Let $F : \mathbb{H} \to \mathbb{C}$ be a holomorphic function where $F(it)$ is real for all $t > 0$. Assume that Serre derivative $\partial_k F$ is positive on the imaginary axis. If $F(it)$ is positive for sufficiently large $t$, then $F(it)$ is positive for all $t > 0$.

A finite-field approximation lemma. If $A_1$ and $A_2$ each have density at least $\alpha$, then for any test set $S$ and $0<\varepsilon\le1$ there is a subspace $V$ of explicitly bounded codimension such that smoothing $\mu_{A_1}*\mu_{A_2}$ by the uniform measure on $V$ changes its total mass on $S$ by at most $\varepsilon$.