Result: On the Kronecker-Hardy-Littlewood theorem

We give a relatively short, elementary proof of the following theorem of \textit{G. H. Hardy} and \textit{J. E. Littlewood} [Acta Math. 37, 155-191 (1914)]: ''For every integer \(r\geq 1\) and for every irratioal \(\theta\), the sequence \((n^ r\theta)_{n\geq 1}\) is everywhere dense (mod 1)''. This generalizes the well-known (but much easier!) theorem of Kronecker, which is the special case \(r=1\) in the preceding theorem. The main improvement in the proof is the use of a lemma of \textit{H. Furstenberg} [Math. Syst. Theory 1, 1-49 (1967; Zbl 0146.285)], which asserts that under sufficiently general conditions a sequence \((x_ n)_{n\geq 1}\) which has 0 as a limit point is everywhere dense (mod 1). Another simple proof based on a different idea can be found in \textit{H. Furstenberg}'s book [Recurrence in ergodic theory and combinatorial number theory (1981; Zbl 0459.28023); Theorems 1.4, 1.6 and 1.26].