We have

Theorem. Let $\psi(x)$ and $\varphi(x)$ be positive increasing functions such that $$\int_1^\infty \frac{dx}{\psi(x)}=+\infty,\qquad \int_1^\infty \frac{dx}{\varphi(x)}<+\infty.$$ Then for almost all $\alpha\in(0,1)$ we have $$\Omega(\log N\cdot \psi(\log\log N))\le\sup_{n\le N}\sum_{j=1}^n(-1)^{\lfloor j\alpha\rfloor}=O(\log N\cdot \varphi(\log\log N)).$$

This is proved in my paper with Jan van de Lune On Some oscillating sums, Uniform Distribution Theory 3 (2008) 35--72.

In this paper it is contained an algorithm to compute the sums. We obtain for example $$S_{\sqrt{2}}(10^{1000})=-10,\quad S_{\sqrt{2}}(10^{10000})=166,\quad S_{\pi}(10^{10000})=11726.$$ With $S_\alpha(N)=\sum_{j=1}^n(-1)^{\lfloor j\alpha\rfloor}$.

We have

Theorem. Let $\psi(x)$ and $\varphi(x)$ be positive increasing functions such that $$\int_1^\infty \frac{dx}{\psi(x)}=+\infty,\qquad \int_1^\infty \frac{dx}{\varphi(x)}<+\infty.$$ Then for almost all $\alpha\in(0,1)$ we have $$\Omega(\log N\cdot \psi(\log\log N))\le\sup_{n\le N}\sum_{j=1}^n(-1)^{\lfloor j\alpha\rfloor}=O(\log N\cdot \varphi(\log\log N)).$$

This is proved in my paper with Jan van de Lune On Some oscillating sums, Uniform Distribution Theory 3 (2008) 35--72.

In this paper it is contained an algorithm to compute the sums. We obtain for example $$S_{\sqrt{2}}(10^{1000})=-10,\quad S_{\sqrt{2}}(10^{10000})=166,\quad S_{\pi}(10^{10000})=11726.$$ With $S_\alpha(N)=\sum_{j=1}^n(-1)^{\lfloor j\alpha\rfloor}$.