Let \(K\) be an algebraic number field of degree \(k\), and let \(\alpha_1,\ldots,\alpha_n\) be linearly independent elements of \(K\) over \(\mathbb Q\). Let \(m\) be a nonzero integer and consider the norm form equation \[ N_{K/\mathbb Q}(x_1\alpha_1+\dots+x_n\alpha_n)=m \] in integers \(x_1,\ldots,x_n\). Let \(H\) denote the solution set of the equation above. Several arithmetic properties of the elements of \(H\) were studied by Everest, Győry, Mignotte and Shorey.
In this paper the authors are concerned with arithmetic progressions in \(H\). Arrange the elements of \(H\) in an \(|H|\times n\) array \(\mathcal H\). Two natural questions appeared about arithmetical progressions. The “horizontal” one: do there infinitely many rows of \(H\) exist which form arithmetic progressions; and “vertical” one: do arbitrary long arithmetic progressions in some column of \(\mathcal H\) exist?
The “horizontal” problem was treated by Bérczes, Pethő and later Bérczes, Pethő, Ziegler, Bazsó computed horizontal solutions. The main goal of this paper is to generalize the result of Pethő and Ziegler to arbitrary norm form equations.
The authors prove that the length of any arithmetic progressions in \(H\) is bounded. Their next theorem shows that in general if \(H\) contains algebraic arithmetic progressions at all, then it contains infinitely many. Moreover, they generalize the following result (Newman): the length of arithmetic progressions consisting of units of an algebraic number field of degree \(k\) is at most \(k\).