A Generalization of a Result of Hardy and Littlewood

TL;DR

This paper explores the growth of a sum involving the fractional parts of multiples of a real number, extending Hardy and Littlewood's results by providing a simple proof and analyzing behavior for generic lpha with respect to certain growth functions.

Contribution

It offers a straightforward proof of Hardy and Littlewood's result for numbers of bounded type and characterizes the sum's growth for generic lpha using divergence and convergence of series.

Findings

Sum is approximately M log M for bounded type lpha.

Growth behavior for generic lpha depends on series convergence.

Provides criteria for the sum's limsup based on lpha's properties.

Abstract

In this note we study the growth of \sum_{m=1}^M\frac1{\|m\alpha\|} as a function of M for different classes of \alpha\in[0,1). Hardy and Littlewood showed that for numbers of bounded type, the sum is \simeq M\log M. We give a very simple proof for it. Further we show the following for generic \alpha. For a non-decreasing function \phi tending to infinity, \limsup_{M\to\infty}\frac1{\phi(\log M)}\bigg[\frac1{M\log M}\sum_{m=1}^M\frac1{\|m\alpha\|}\bigg] is zero or infinity according as \sum\frac1{k\phi(k)} converges or diverges.