Research proof index

For every $\delta,\varepsilon>0$, there is an explicit positive uniform bound $C(\delta,\varepsilon)$ such that, if a measurable $2\pi$-periodic function $g$ satisfies $\lVert g(x)\rVert\le C(\delta,\varepsilon)$ for every $x$, then the set where $\sup_N\lVert S_Ng(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$.

If every tile in $P$ lies at the fixed generation $k$, then the first global density of $P$ is at most the generation-specific density $\mathrm{dens}'(k,P)$.

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

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

The project’s structural constants are chosen so that the scale-separation factor has the fixed quantitative bound

$D^{-\kappa Z}\le2^{-100}.$

The operator czOperator K r is bounded from L^2 ([1, 4]) to L^2 ([2, 3]), uniformly in r. This follows from the fact, proved in norm_czOperator_le_add, that it is bounded by the sum of two operators which are both bounded: one is the convolution with dirichletApprox, bounded as it is an average of Fourier projections, and the other one has a k...