Towards a sharp converse of Wall's theorem on arithmetic progressions (Q2305152)

The author of the present article investigates Wall's theorem on arithmetic progressions of digits. In particular, he shows a kind of converse statement. The used methods are of elementary nature. Let \(b\geq2\) be an integer. Suppose that \(x\in[0,1)\) hase base \(b\)-expansion \(x=0.a_1a_2a_3\ldots\) with \(a_i\in\{0,1,\ldots,b-1\}\) for \(i\geq 1\). Then we call \(x\) normal (in base \(b\)) if for every integer \(k\geq1\) and every finite string \(s=[d_1,d_2,\ldots,d_k]\) with \(d_i\in\{0,1,\ldots,k-1\}\) for \(1\leq i\leq k\) the asymptotic frequency of occurrences of this string in the base \(b\)-expansion is \(b^{-k}\), i.e., \[\lim_{n\to\infty}\frac{\#\{0\leq i\leq n-1\colon a_{i+j}=d_j,j=1,2,\ldots,k\}}{n}=\frac{1}{b^k}.\] Then Wall's theorem states that if \(0.a_1a_2a_3\) is normal (in base \(b\)) then for any \(k,\ell\in\mathbb{N}\) the number \(0.a_{k}a_{k+\ell}a_{k+2\ell}\ldots\) is also normal (in base \(b\)). The author's first observation is that the exclusions of the trivial case \(\ell=1\) is not sufficient. In particular, he shows the existence of numbers \(0.a_1a_2a_3\ldots\in[0,1)\) which are not normal (in base \(b\)) such that for every \(k\in\mathbb{N}\) and every \(\ell\in\mathbb{N},\ell\geq2\), the number \(0.a_{k}a_{k+\ell}a_{k+2\ell}\ldots\) is normal. Let \(x=0.a_1a_2a_3\ldots\in[0,1)\) and suppose that for any \(\varepsilon>0\) there exists an increasing sequence \(n_1<n_2<n_3<\cdots\) of positive integers with asymptotic lower density greater than \(1-\varepsilon\) such that \(0.a_{n_1}a_{n_2}a_{n_3}\ldots\in[0,1)\) is normal (in base \(b\)). Then the author shows that \(x\) is normal (in base \(b\)). Finally, let \(\mathcal{N}\) be a collection of periodic increasing sequences \(n_1<n_2<n_3<\cdots\) of the positive integers. Suppose there exist elements \(K\) and \(L\) of \(\mathbb{N}\) such that \[\{K(n-1)+L,K(n-1)+L+1,K(n-1)+L+2,\ldots,Kn+L-1\}\] is not a subset of any of the sequences for any \(n\in\mathbb{N}\). Then the author shows that there exists a real number \(0.a_1a_2a_3\ldots\in[0,1)\) that is not normal (in base \(b\)), but \(0.a_{n_1}a_{n_2}a_{n_3}\ldots\in[0,1)\) is normal (in base \(b\)) for all sequences in \(\mathcal{N}\).