The size of Max\((p)\) sets and density bases.
Consider a subset \(A\) of the unit circle in \(\mathbb R^2\), and view \(A\) as a set of directions. Given \(x\in \mathbb R^2\), let \(A_x\) be the set of all rectangles containing \(x\) and oriented in one of the directions of \(A\). If it is true that \(\lim_{R\in A_x,{\roman{diam}}\, R\to 0}| E\cap R| /| R| =\chi_{E}(x)\) a.e., \(A\) is said to be a density set. The Hardy-Littlewood maximal operator associated with the set of directions \(A\) is defined as \(M_A(f)(x)=\sup_{R\in A_x}| R| ^{-1}\int_R | f| \). When \(M_A\) maps \(L^p(\mathbb R^2)\) to \(L^p(\mathbb R^2)\) for some \(1<p<\infty\), \(A\) is called a \({\roman{Max}}(p)\) set. A set of real numbers \(\{x_j\}_{j=1}^{\ell}\) is called an almost arithmetic progression of step \(d\) and length \(\ell\) if there is a number \(a\) such that \(a+(j-1)d\leq x_j<a+jd\) for all \(j=1,\ldots,\ell\). The main theorem in this paper is: Suppose \(A\subset [0,2\pi)\) is a \({\roman{Max}}(p)\) set for some \(1<p<\infty\) or a density basis and assume \(f\) is a function satisfying \(f(x)=o(\log x)\). Then there exists an integer \(M_0\) depending on \(A\) and \(f\) such that both \(A\) and the set of cotangents of the angles in \(A\) contain at most \(2^M-f(2^M)\) terms from any almost arithmetic progression of length \(2^M\), \(M\geq M_0\). From this the authors derive several results concerning the thickness of sets. For example, they give the following: If the Hausdorff dimension of the compact set \(E\) is one, then \(E\) is neither a \({\roman{Max}}(p)\) set for any \(1<p<\infty\) nor a density set.
scientific article; zbMATH DE number 1890481