This is studied in detail in Sums of Fractional Parts of Integer Multiples of an Irrational
The sum $$C(N)=\sum_{n=1}^{N} ( \{ n \xi \} - \frac{1}{2})$$ satisfies $|C(N)|>c\log N$ for infinitely many $N$. The coefficient $c\geq 1/720$.

This is studied in detail in Sums of Fractional Parts of Integer Multiples of an Irrational
The sum $$C(N)=\sum_{n=1}^{N} \bigl( \{ n \xi \} - \tfrac{1}{2}\bigr)$$ satisfies $|C(N)|>c\log N$ for infinitely many $N$. The coefficient $c\geq 1/720$.

This is studied in detail in Sums of Fractional Parts of Integer Multiples of an Irrational The sum $$C(N)=\sum_{n=1}^{N} ( \{ n \xi \} - \frac{1}{2})$$ satisfies $|C(N)|>c\log N$ for infinitely many $N$. The coefficient $c\geq 1/720$.

This is studied in detail in Sums of Fractional Parts of Integer Multiples of an Irrational
The sum $$C(N)=\sum_{n=1}^{N} ( \{ n \xi \} - \frac{1}{2})$$ satisfies $|C(N)|>c\log N$ for infinitely many $N$. The coefficient $c\geq 1/720$.