On some variants of the Kakeya problem
The authors give the following two results: A Borel set in \(\mathbb R^d\), \(d\geq 3\), which contains a sphere of every radius has Hausdorff dimension \(d\). A Borel set in \(\mathbb R^2\) which contains a circle of every radius has Hausdorff dimension at least \(11/6\). They derive these by showing the following estimates for a related maximal function. Let \[ C_\delta(x,r)=\{z\in\mathbb R^d;\;r-\delta<|z-x|<r+\delta\} \] and \[ M_\delta f(x)=\sup_{x}|C_\delta(x,r)|^{-1}\int_{C_\delta(x,r)}|f(y)|dy. \] Then \[ \|M_\delta f\|_{L^2(1/2,2)}\leq C(\log(1/\delta)^{1/2}\|f\|_2 \] when \(d\geq 3\), and \[ \|M_\delta f\|_{L^q(1/2,2)}\leq C_{p,q}\delta^{-1/2(3/p-1)}\|f\|_p \] for \(1<p<8/3\), \(q\leq 2p'\) when \(d=2\). As a related result, they show that any cylindrically symmetric Borel set in \(\mathbb R^3\) containing a line segment in every direction has Hausdorff dimension at least \(17/6\). To prove this , they give an estimate for a maximal function similar to the above.
scientific article; zbMATH DE number 1463699