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The sequence $\sum a_n$ is unbounded.

This is a consequence of a general result from Kesten,
On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arithmetica (1966). The proof is not very long but quite computational, using properties of continued fractions.

Theorem Let $\xi \in [0,1]$, $0\leq a < b \leq 1$, denote by $N(M, \xi, a,b)$ the number of integers less than $M$ such that $a \leq \{k\xi\} \leq b$. Then $N(M,\xi, a,b) - M(b-a)$ is unbounded if $b-a$ is rational and $\xi$ is irrational.

You are interested in the $1/2$ discrepancy of an irrational rotation of the circle. Note that from the Denjoy-Koksma inequality, there is a sequence $q_n$ such that $\sum_{k= 0}^{q_n-1} a_n$ is bounded by $2$. The sequence exists for any irrational rotation of the circle and is given by the denominators of the partial fraction decomposition of the rotation number. For the golden ratio, these numbers are given by the Fibonacci numbers.

The golden ratio is a bit special because its decomposition in continued fraction is just $[1,1,1...]$ so it is badly approximated by the rational numbers. So the computation of Kesten should be simpler in that case.

EDIT: the theorem is also mentioned shortly in the book of Kuipers and Niederreiter, "uniform distribution of sequences". Some additional references can be found in the notes of chapter 2, section 3 p128.

The sequence $\sum a_n$ is unbounded.

This is a consequence of a general result from Kesten,
On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arithmetica (1966). The proof is not very long but quite computational, using properties of continued fractions.

Theorem Let $\xi \in [0,1]$, $0\leq a < b \leq 1$, denote by $N(M, \xi, a,b)$ the number of integers less than $M$ such that $a \leq \{k\xi\} \leq b$. Then $N(M,\xi, a,b) - M(b-a)$ is unbounded if $b-a$ is rational and $\xi$ is irrational.

You are interested in the $1/2$ discrepancy of an irrational rotation of the circle. Note that from the Denjoy-Koksma inequality, there is a sequence $q_n$ such that $\sum_{k= 0}^{q_n-1} a_k$ is bounded by $2$. The sequence exists for any irrational rotation of the circle and is given by the denominators of the partial fraction decomposition of the rotation number. For the golden ratio, these numbers are given by the Fibonacci numbers.

The golden ratio is a bit special because its decomposition in continued fraction is just $[1,1,1...]$ so it is badly approximated by the rational numbers. So the computation of Kesten should be simpler in that case.

EDIT: the theorem is also mentioned shortly in the book of Kuipers and Niederreiter, "uniform distribution of sequences". Some additional references can be found in the notes of chapter 2, section 3 p128.

The sequence $\sum a_n$ is unbounded.

This is a consequence of a general result from Kesten,
On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arithmetica (1966). The proof is not very long but quite computational, using properties of continued fractions.

Theorem Let $\xi \in [0,1]$, $0\leq a < b \leq 1$, denote by $N(M, \xi, a,b)$ the number of integers less than $M$ such that $a \leq \{k\xi\} \leq b$. Then $N(M,\xi, a,b) - M(b-a)$ is unbounded if $b-a$ is rational and $\xi$ is irrational.

You are interested in the $1/2$ discrepancy of an irrational rotation of the circle. Note that from the Denjoy-Koksma inequality, there is a sequence $q_n$ such that $\sum_{k= 0}^{q_n-1} a_n$ is bounded by $2$. The sequence exists for any irrational rotation of the circle and is given by the denominators of the partial fraction decomposition of the rotation number. For the golden ratio, these numbers are given by the Fibonacci numbers.

The golden ratio is a bit special because its decomposition in continued fraction is just $[1,1,1...]$ so it is badly approximated by the rational numbers. So the computation of Kesten should be simpler in that case.

The sequence $\sum a_n$ is unbounded.

This is a consequence of a general result from Kesten,
On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arithmetica (1966). The proof is not very long but quite computational, using properties of continued fractions.

Theorem Let $\xi \in [0,1]$, $0\leq a < b \leq 1$, denote by $N(M, \xi, a,b)$ the number of integers less than $M$ such that $a \leq \{k\xi\} \leq b$. Then $N(M,\xi, a,b) - M(b-a)$ is unbounded if $b-a$ is rational and $\xi$ is irrational.

You are interested in the $1/2$ discrepancy of an irrational rotation of the circle. Note that from the Denjoy-Koksma inequality, there is a sequence $q_n$ such that $\sum_{k= 0}^{q_n-1} a_n$ is bounded by $2$. The sequence exists for any irrational rotation of the circle and is given by the denominators of the partial fraction decomposition of the rotation number. For the golden ratio, these numbers are given by the Fibonacci numbers.

The golden ratio is a bit special because its decomposition in continued fraction is just $[1,1,1...]$ so it is badly approximated by the rational numbers. So the computation of Kesten should be simpler in that case.

EDIT: the theorem is also mentioned shortly in the book of Kuipers and Niederreiter, "uniform distribution of sequences". Some additional references can be found in the notes of chapter 2, section 3 p128.

The sequence $\sum a_n$ is unbounded.

This is a consequence of a general result from Kesten,
On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arithmetica (1966). The proof is not very long but quite computational, using properties of continued fractions.

Theorem Let $\xi \in [0,1]$, $0\leq a < b \leq 1$, denote by $N(M, \xi, a,b)$ the number of integers less than $M$ such that $a \leq \{k\xi\} \leq b$. Then $N(M,\xi, a,b) - M(b-a)$ is unbounded if $b-a$ is rational and $\xi$ is irrational.

You are interested in the $1/2$ discrepancy of an irrational rotation of the circle. Note that from the Denjoy-Koksma inequality, there is a sequence $q_n$ such that $\sum_{k= 0}^{q_n-1} a_n$ is bounded by 2. The sequence exists for any irrational rotation of the circle and is given by the denominators of the partial fraction decomposition of the rotation number. For the golden ratio, these numbers are given by the Fibonacci numbers.

The golden ratio is a bit special because its decomposition in continued fraction is just [1,1,1...] so it is badly approximated by the rational numbers. So the computation of Kesten should be simpler in that case.

The sequence $\sum a_n$ is unbounded.

This is a consequence of a general result from Kesten,
On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arithmetica (1966). The proof is not very long but quite computational, using properties of continued fractions.

Theorem Let $\xi \in [0,1]$, $0\leq a < b \leq 1$, denote by $N(M, \xi, a,b)$ the number of integers less than $M$ such that $a \leq \{k\xi\} \leq b$. Then $N(M,\xi, a,b) - M(b-a)$ is unbounded if $b-a$ is rational and $\xi$ is irrational.

You are interested in the $1/2$ discrepancy of an irrational rotation of the circle. Note that from the Denjoy-Koksma inequality, there is a sequence $q_n$ such that $\sum_{k= 0}^{q_n-1} a_n$ is bounded by $2$. The sequence exists for any irrational rotation of the circle and is given by the denominators of the partial fraction decomposition of the rotation number. For the golden ratio, these numbers are given by the Fibonacci numbers.

The golden ratio is a bit special because its decomposition in continued fraction is just $[1,1,1...]$ so it is badly approximated by the rational numbers. So the computation of Kesten should be simpler in that case.