A CURIOSITY ABOUT (−1)[e] + (−1)[2e] + · · · + (−1)[Ne]

Let \( \alpha \) be an irrational real number; the behaviour of the sum

\[
S_N(\alpha) := (-1)^{\lfloor \alpha \rfloor} + (-1)^{\lfloor 2\alpha \rfloor} + \cdots + (-1)^{\lfloor N\alpha \rfloor}
\]

depends on the continued fraction expansion of \( \alpha/2 \). Since the continued fraction expansion of \( \sqrt{2}/2 \) has bounded partial quotients, \( S_N(\sqrt{2}) = O(\log(N)) \) and this bound is best possible. The partial quotients of the continued fraction expansion of \( e \) grow slowly and thus \( S_N(2e) = O\!\left(\dfrac{\log(N)^2}{\log\log(N)^2}\right) \), again best possible. The partial quotients of the continued fraction expansion of \( e/2 \) behave similarly as those of \( e \). Surprisingly enough \( S_N(e) = O\!\left(\dfrac{\log(N)}{\log\log(N)}\right) \).