Research proof index

If a finite set $s$ is additively dissociated, then its order-$n$ additive energy is at most $C^n n^n |s|^n$, where $C$ is the project's Chang constant. This is the quantitative dissociated-set estimate used in the proof of Chang's lemma.

adjointCarleson is the adjoint of carlesonOn.

Suppose that, for every error threshold $\delta>0$ and every measure tolerance $\varepsilon>0$, one can choose $N_0$ so that the set where $\sup_{N>N_0}\lVert f(x)-F_N(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$. Then $F_N(x)$ converges to $f(x)$ for almost every $x$.

An $L^\infty$ almost-periodicity theorem. Under the small-growth hypothesis $\sigma[A,S]\le K$, and for nonempty finite sets $B,C$, there is a set of translations $T$ with $|T|\ge K^{-4096\lceil\mathcal L(|C|/|B|)\rceil/\varepsilon^2}|S|$. Every $t\in T$ changes the normalized convolution $\mu_A*1_B*\mu_C$ by at most $\varepsilon$ in $L^\infty$.

A boosted $L^\infty$ almost-periodicity estimate. Under $\sigma[A,S]\le K$, it finds a large set $T$, with the stated lower bound $|T|\ge K^{-4096\lceil\mathcal L(|C|/|B|)\rceil k^2/\varepsilon^2}|S|$, such that averaging the target convolution against the $k$-fold convolution of $\mu_T$ changes it by at most $\varepsilon$ in $L^\infty$.

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

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