Using large prime divisors to construct normal numbers

Abstract. Given an integer \(q\ge 2\), a \(q\)-normal number is an irrational number \(\xi\) such that any preassigned sequence of \(\ell\) digits occurs in the \(q\)-ary expansion of \(\xi\) at the expected frequency, namely \(1/q^\ell\). Let \(\eta(x)\) be a slowly increasing function such that \(\frac{\log \eta(x)}{\log x} \to 0\) as \(x\to \infty\). Then, letting \(P(n)\) stand for the largest prime factor of \(n\), set \(Q(n)\) to be the smallest prime divisor of \(n\) which is larger than \(\eta(n)\), while setting \(Q(n)=1\) if \(P(n)> \eta(n)\). Then, we show that the real number \(0.Q(1)Q(2)\ldots\) is a normal number in base 10. With various similar constructions, we create large families of normal numbers in any given base \(q\ge 2\). Finally, we consider exponential sums involving the \(Q(n)\) function.