The fact that the variance of the sum of independent random variables is the sum of their variances allows one to have a good understanding of how well-concentrated each term $X\_i$ in a sum of $n$ independent random variables $S= X\_1+X\_2+\dots+X\_n$ has to be in order for $S$ to be concentraded around the mean. It seems that an analogous fact should be true for the sum taken modulo $1$. More concretely, I believe that the following should be true, but I cannot seem to find a proof or a reference. Help finding either of these would be highly appreciated.


**Lemma (tentative).** Suppose that $X\_1, X\_2, ..., X\_n$ are (jointly) independent real random variables with $\mathbb{E} X\_i = 0$ for each $i \leq n$, and that there exists a real number $\gamma$ such that 
$$ \mathbb{E} [ \lVert X\_1 + X\_2 + \dots + X\_n - \gamma \rVert\_{\mathbb{R}/\mathbb{Z}}^2] \leq \varepsilon $$
for some small $\varepsilon > 0$. Then 
$$ \mathbb{E} [ \lVert X\_1\rVert\_{\mathbb{R}/\mathbb{Z}}^2 ] + \mathbb{E} [ \lVert X\_2\rVert\_{\mathbb{R}/\mathbb{Z}} ^2 ] + \dots + \mathbb{E} [ \lVert X\_n\rVert\_{\mathbb{R}/\mathbb{Z}}^2 ] \leq C \varepsilon,$$
where $C$ is an absolute constant. Here, $\lVert x\rVert\_{\mathbb{R}/\mathbb{Z}} $ denotes the distance of $x$ from the nearest integer.


*Rationale:* 


1. The lemma would be true if $\lVert x\rVert\_{\mathbb{R}/\mathbb{Z}} $ were replaced with $\lvert x\rvert\_{\mathbb{R}/\mathbb{Z}} $; indeed this is just the additivity of variance.
2. It is not hard to check that the lamma is true if $X\_i$ have normal distribution or if $X\_i$ have Bernoulli distribution with vales $0$ and $1/2$.
3. On the intuitive level, if we imagine that the distribution of $X\_1+\dots+X\_n$ is roughly bell-shaped with a single maximum, the only way for $\mathbb{E} [ \lVert X\_1 + X\_2 + \dots + X\_n - \gamma \rVert\_{\mathbb{R}/\mathbb{Z}}^2]$ to be small is if $X\_1 + X\_2 + \dots + X\_n$ is very strongly concentrated around that maximum. In particular the ``wrap-around'' issues should not play much role and the above lemma should be roughly equivalent to the version with absolute value in place of the distance from $\mathbb{Z}$.



Variance modulo 1
On the one hand, the proof is very cheap. Let $Z\_j=e^{2\pi iX\_j}$. $X=\sum\_j X\_j$, $Z=e^{2\pi i X}$. Note that $\operatorname{Var}\_{\mathbb R/\mathbb Z}X\approx 1-|EZ|$ and similarly for $X\_j$ and $Z\_j$. Now just use the identity $EZ=\prod\_j EZ\_j$ to conclude.


On the other hand, finding the reference may be a highly non-trivial task, so I leave it to somebody else :-).


P.S. What can be really concluded here is that there exist $\gamma\_j$ summing to $\gamma$ with $\sum\_jE\|X\_j-\gamma\_j\|\_{\mathbb R/\mathbb Z}^2\le C\varepsilon$. The condition $EX\_j=0$ does not allow one to conclude from here that we can take $\gamma\_j=0$: take $X\_j$ symmetric with values about $\pm \frac 12$ for a counterexample.


