 Solution

Independence gives $S_n\sim N(0,n)$. For every integer $m$,since subtraction of an integer does not change the exponential. These are precisely the Fourier coefficients of the distribution on the circle.

One can also obtain a direct density proof on $[0,1]$, avoiding an endpoint issue in transferring circle convergence. Periodize the normal density:Its coefficients are the displayed Gaussian values, so its absolutely convergent Fourier series isThe sum of the absolute values of the nonconstant coefficients tends to zero, for example by domination by the summable sequence at $n=1$. Therefore $f_n\to 1$ uniformly, and integrating any bounded continuous function on $[0,1]$ givesIn fact the convergence is in total variation.