We consider certain Birkhoff Sums, namely

$$S_n(x):=\sum_{i=1}^n \{x+i\rho\}$$

These are connected to various subbranches of mathematics: numerical analysis, number theory, ergodic theory, and dynamical systems.

* The discrepancy of a sequence $\{x_i\}_{i=1}^\infty$ in $[0,1)$ is a measure of how unevenly $\{x_i\}_{i=1}^n$ is distributed for each $n$ (Pisot and Van Der Corput in the 1930s). We show that the discrepancy of $\{i\rho\}_{i=1}^n$ equals the length of the range of $S_n(x)$.

* If $q_n$ is a continued fraction denominator of $\rho$, we compute the discrepancy of $\{i\rho\}_{i=1}^{q_n}$ exactly.

* We show how to compute exactly $S_n(0)$ for all $n$ and conjecture exact growth rates of $S_n(0)$ for rotations by `metallic means' (or $\rho=[a,a,\cdots]$). We prove this for $a=1$ and $2$.

* Technical details are suppressed to keep the exposition accessible for graduate students.