Return to Revisions

No $x$ less then $1/2$ may satisfy $S_N=o(n^x)$. Indeed, denote $f(x)=\chi_{[0,\pi]}-\chi_{[\pi,2\pi]}$, then we are interested in $|f(\theta)+f(2\theta)+\dots+f(n\theta)|$ for specific value of $\theta$. But $$\int_0^{2\pi} |f(\theta)+f(2\theta)+\dots+f(n\theta)|^2 d\theta$$ is not less then $n$, since $\int f^2(k\theta)=1$, $\int f(k\theta) f(m\theta)\geq 0$ (the latter may be otten elementary or via Fourier series $f(x)=\pi^{-1}\sum \sin (2k+1)x/(2k+1)$, I may be wrong with the constant $\pi^{-1}$).