Powers in arithmetic progressions
The authors analyse the function \(P_{a,b;N}(l)\), for integers \(a,b,l\) with \(a>0,l\geq 2\), which enumerates the \(l^{th}\)-powers among the first \(N\) terms \(b,b+a,\cdots,b+(N-1)a\) of the arithmetic progression \(ax+b, x\geq 0\). If \(P_N(l)\) denotes the maximum of these values taken over all arithmetic progressions \(ax+b\), then a conjecture of W.~Rudin predicts that \(P_N(2)=O(\sqrt{N})\). Let \[ S_{a,b}(l)=\underset{N\rightarrow\infty}{\lim}\frac{\mid \{x:ax+b \text{ is an }l^{th} \text{power },0\leq x<N\}\mid }{\sqrt[l]{N}}. \] Let \(S_a(l)=\underset{b\in \mathbb{Z}}{\max}\ S_{a,b}(l)\) and \(S(l)=\underset{a\in \mathbb{N}}{\max}\ S_a(l)\). The \(l\)-transformation of a progression \(ax+b\) is defined as the arithmetic progression \((az^l)x+(b+ta)z^l\) for some positive integer \(z\) and \(t\) an integer. They prove the following main theorem in the article. \(S(l)\) exists for any \(l\geq 2\) and we have \[ S(l)=\begin{cases} \sqrt{\frac 83} & \text{ if }l=2,\\ \underset{\overset{p\text{ prime, } p-1\mid l}{\frac{\log\ p}{\log\ p-\log(p-1)}>l}}{\prod} (p-1)p^{\frac 1l-1} & \text{ otherwise.} \end{cases} \] Further, for the arithmetic progression \(ax+b\) we have \(S_{a,b}(l)=S(l)\) if and only if it is an \(l\)-transformation of \(a^*x+b^*\) where \(a^*=2\) if \(l=2\), \(a^*=5\) or \(80\) if \(l=4\), \(a^*=\underset{\overset{p\text{ prime, } p-1\mid l}{\frac{\log\ p}{\log\ p-\log(p-1)}>l}}{\prod} p\) otherwise and \(b^*=0\) if \(a^*=1\), \(b^*=1\) otherwise. Based on the main theorem, they suggest two problems in the article which are extensions of Rudin's conjecture to the case of general exponents \(l\). It is mentioned that the answer to these two questions proposed is affirmative. To support this, they provide numerical results for \(l=3\) up to \(N<20\) and for \(l=4\) up to \(N<6\) using elliptic and higher genus curves. The affirmative answers to the two proposed questions would give that \(\underset{a>0,b\in \mathbb{Z}}{\max}\ P_{a,b;N}(l)=O(\sqrt[l]{N})\) which provides a positive answer to the Rudin's conjecture, extended to any exponent \(l\).
scientific article; zbMATH DE number 7383300
scientific article