How many intervals cover a point in random dyadic covering? (Q5953291)

Let \({\mathbb D}\) denote the set of \(\{0,1\}\)-sequences. For \(k\in{\mathbb N}\) and \((t_1,\dots,t_k)\in \{0,1\}^k\), let \(X\sim X_{t_1,\dots,t_k}\) be i.i.d.\ non-negative integer-valued random variables. Then the \(n\)-covering number of \(t\in{\mathbb D}\) is defined as the random variable \(N_n(t):=\sum_{k=1}^nX_{t_1,\dots,t_k}\). The main aim of the paper is to study the size of the random sets \(E_{b,s}:=\{t\in{\mathbb D}: N_n(t)-bn\sim s_n\, (n\to\infty)\}\), where \(b \in\mathbb R\) and \(s=(s_n)\) is a sequence of positive real numbers satisfying some growth conditions. For instance, a formula for the Hausdorff and the packing dimension of these sets is given, if \(b\) is taken from a suitable range and the moment generating function of \(X\) is finite.