How many intervals cover a point in Dvoretzky covering?
Let the circle \(T=R/Z\) be identified with the interval \([0,1)\). Let \(\{l_n\}\) \((n\geq 1, 0<l_n<1)\) be a decreasing sequence such that \(\sum_nl_n=\infty\), and let \(\{\omega_n\}\) \((n\geq 1\)) be an i.i.d. random sequence of the uniform distribution. Denote by \(I_n\) the open interval of length \(l_n\), with left end point \(\omega_n\). The Dvoretzky covering problem is to give necessary conditions and sufficient conditions on the length of the sequence \(\{l_n\}\) in order to cover the whole circle \(T\) almost surely. This is equivalent to \(P(T= \bigcup^\infty_{n=1} I_n)=1\), \(P\) being the probability measure of the underlying probability space \((\Omega,\mathcal{A},P)\). A complete solution to the problem was given by \textit{L. A. Shepp} [Isr. J. Math. 11, 328-345 (1972; Zbl 0241.60008)]. He showed that \(\bigcup^\infty_{n=1} I_n\) almost surely covers \(T\) precisely when \(\sum^\infty_{n=1} (1/n^2)\exp (l_1+\cdots +l_n)=\infty\); further, under this condition, each point of \(T\) is almost surely covered by infinitely many intervals. For \(t\in T\), let \(N_n(t)\) be the number of intervals in \(\{I_1,\dots, I_n\}\) which cover \(t\). A natural problem is to investigate \(N_n(t)\) for \(n\to\infty\) for a given length sequence. In a previous paper [Ann. Inst. Henri Poincaré, Probab. Stat. 29, No. 3, 453-466 (1993; Zbl 0799.60013)] the author and \textit{J.-P. Kahane} proved that, for \(l_n=\alpha/n\), \(\alpha>1\), there exist two constants \(A_{\alpha},B_{\alpha}\), with \(0<A_{\alpha}\leq\alpha\leq B_{\alpha}<\infty\) such that \[ A_{\alpha}\leq\liminf_{n\to\infty}N_n(t)/\log n\leq\limsup_{n\to\infty}N_n(t)/\log n\leq B_{\alpha} \text{ a.s. }\forall t\in T. \] From this it derives that \(\lim_{n\to\infty}N_n(t)/\log n=\alpha\) a.s. and for almost every \(t\in T\) (with respect to Lebesgue measure). Let \(l_n=\alpha/n\), \(\alpha>0\), and let \(\beta\neq\alpha\) be a positive number, and let \(F_{\beta}=\{t\in T:\lim_{n\to\infty}N_n(t)/\log n=\beta\}\). Denote by \(\dim F_{\beta}\) the Hausdorff dimension of \(F_{\beta}\). The author proves the following interesting properties: (1) If \(\beta\in(\alpha-\sqrt{\alpha},\alpha+\sqrt{\alpha})\cap (0,\infty)\), then a.s. \(\dim F_{\beta}\geq 1-|\beta-\alpha|^2/\alpha>0\). (2) If \(\beta\log(\beta/\alpha)-(\beta-\alpha)\leq 1\), then a.s. \(\dim F_{\beta}\leq 1-[\beta\log(\beta/\alpha)-(\beta-\alpha)]\) (this holds for the packing dimension of \(F_{\beta}\)). (3) If \(\max(\alpha-1,0)<\beta<\alpha\), then a.s. \(\dim F_{\beta}=1-[\beta\log(\beta/\alpha)-(\beta-\alpha)]\). (4) The set of \(t\in T\) such that \(\liminf_{n\to\infty}N_n(t)/\log n<\limsup_{n\to\infty}N_n(t)/\log n\) is a.s. of Hausdorff dimension \(1\). In particular, from (3), it follows that points on the circle are differently covered.
scientific article; zbMATH DE number 1850608
scientific article