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Let ũ denote the conjugate Poisson integral of a function $f\in L^p\left(ℝ\right)$. We give conditions on a region Ω so that $li{m}_{(v,\epsilon )\to (0,0)}(v,\epsilon )\in \Omega ũ(x+v,\epsilon )=Hf\left(x\right)$, the Hilbert transform of f at x, for a.e. x. We also consider more general Calderón-Zygmund singular integrals and give conditions on a set Ω so that $su{p}_{(v,r)\in \Omega }\left|{ʃ}_{\left|t\right|>r}k(x+v-t)f\left(t\right)dt\right|$ is a bounded operator on $L^p$, 1 < p < ∞, and is weak (1,1).

A sum of exponentials of the form $f\left(x\right)=exp\left(2\pi i{N}_{1}x\right)+exp\left(2\pi i{N}_{2}x\right)+\cdots +exp\left(2\pi i{N}_{m}x\right)$, where the $N_k$ are distinct integers is called an (because the convolution of $f$ with itself is $f$) or, simply, an . We show that for every $p\&gt;1,$ and every set $E$ of the torus $𝕋=ℝ/\mathbb{Z}$ with $\left|E\right|\&gt;0,$ there are idempotents concentrated on $E$ in the $L^p$ sense. More precisely, for each $p\&gt;1,$ there is an constant $C_p\&gt;0$ so that for each $E$ with $\left|E\right|\&gt;0$ and $ϵ\&gt;0$ one can find an idempotent $f$ such that the ratio ${\left({\int }_E{\left|f\right|}^p/{\int }_{𝕋}{\left|f\right|}^p\right)}^{1/p}$ is greater than $C_p-ϵ$. This is in fact a lower bound result and, though optimal, it is close to the...