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With $\sum_{\nu \ge 0}\phi_\nu(\xi)=1$ be a Littlewood-Paley partition of unity we find that $u=\sum_{\nu \ge 0}\phi_\nu(D)u$ and thus since $$\Vert u\Vert_{B^\alpha_{\infty, \infty}}=\sup_{\nu\in \mathbb N} 2^{\nu \alpha}\Vert\phi_\nu(D)u\Vert_{L^\infty},$$ we get $$\Vert u\Vert_{L^\infty}\le \sum_{\nu \ge 0}2^{\nu \alpha}\Vert\phi_\nu(D)u\Vert_{L^\infty}... 1 Come on, for the first property just use the fact that$$ \int_{-n}^n K(x) dx = \sum_{j = -n+1}^{n-1} \phi(j) + \tfrac12(\phi(-n)+\phi(n)) $$has a finite limit as n \to \infty, together with convergence of$$ \biggl| \int_{-a}^a K(x) dx - \int_{-\lfloor a\rfloor}^{\lfloor a\rfloor} K(x) dx \biggr| \leqslant |\phi(-\lfloor a\rfloor-1)| + |\phi(-\lfloor a\...

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Section 6.1 of Yufei Zhao's notes Graph theory and Additive Combinatorics discusses Roth's theorem in finite fields. The version of the Roth's theorem states that a set $X$ in the vector space $\mathbb{F}_3^n$ without three-term arithmetic progressions (3-AP) has size at most $O(3^n/n)$. The argument is as follows: "Decompose the original object into ...

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