Research proof index

Fix $1<p<2$, an error threshold $\delta>0$, and a measure tolerance $\varepsilon\ge0$. There is an explicit bound $C(\delta,\varepsilon,p)$ such that, whenever a measurable $2\pi$-periodic function $g$ satisfies $\lVert g\rVert_{L^p((0,2\pi])}\le C(\delta,\varepsilon,p)$, the set where the maximal partial Fourier sum $\sup_N\lVert S_Ng(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$.

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 $\mathrm{dens}'(k,P)$.

This is the dual of Dense.comap_val_nhdsWithin_Ioi_neBot.

This is an auxillary lemma used to prove Dense.monotone_of_isRightContinuous. It is saying that if D is a dense set and a, b are two points such that a < b, then the comap of 𝓝[Set.Ioi a] a under the inclusion D → α is nontrivial. Note that a < b is necessary as this is clearly not true if a is a top element.

If f is monotone on a dense set D and is right continuous, then f is monotone. We prove under the assumption that α has a top element ⊤ and ⊤ ∈ D, which is a necessary assumption because otherwise it is possible that ⊤ is an isolated point. This theorem should be also true when α satisfies NoTopOrder α.