TITLE: Sum of Gaussian pdfs
QUESTION [25 upvotes]: I learned from a colleague that if one sums translates of the Gaussian density $f(x)=(2\pi)^{-1/2}e^{-x^2/2}$ translated by the integers (i.e. one considers $F(x)=\sum_{n\in\mathbb Z}f(x+n)$), the resulting function is remarkably constant: that is, the function differs from its average value by less than one part in $10^8$.  The significance of translating by multiples of 1 is that the inflection point of the Gaussian occurs at $\pm1$. 
I attempted to repeat this for $g(x)=e^{-3x^4/4}$ which also has an inflection point at $\pm 1$, and found that the resulting sum is constant to one part in 200. 

Can anyone offer any conceptual explanation for the remarkable degree of constancy of $F(x)$, or is this just a fluke?

REPLY [8 votes]: Interesting! Put another way, the fractional part of the standard Gaussian variable closely approximates the standard uniform variable.
You can also closely approximate standard Gaussian from standard uniform: Add 12 independent uniform variables before subtracting 6. This approximation is also shockingly good; see John D. Cook's comment here. (12 isn't "large," but rather uses var(U)=1/12 to get unit variance.)
Notice that the fractional part of this approximate Gaussian is exactly 
uniform. You might think of this as a dual explanation for the phenomenon you observed.