# nt.number theory – Lower limit for some unit root sums

Leave $$n$$ be a positive integer (assumes $$n$$ is paramount to simplicity), and leaves $$x_k = pm1$$, for $$k = 0,1,2, …, n-1$$. Leave $$rho$$ bean $$n-$$At the root of the unit, I am interested in the lower limits for the absolute value of the sums of the form:

$$S_n = sum_ {k = 0} ^ {n-1} x_k rho ^ k$$ assuming that this sum is not equal to zero (when $$n$$ it is essential that this can only happen when all the $$x_k$$they have the same sign)

One can get an easy lower limit of $$frac {1} {n ^ {n-1}}$$ multiplying this algebraic integer by all its Galois conjugates, but since there are $$2 ^ n$$ sums, I'm waiting for a better lower limit (hopefully $$e ^ {- Cn}$$ for some constant $$C$$), or maybe there is something known about probability $$Pr (| S_n | I hope this amount is exponentially small, I think from an argument in the Tao-Vu article (https://arxiv.org/abs/1307.4357) related to the optimal Offord-Littlewood inverse theorem of Nguyen-Vu, one could show that such a probability is less than $$n ^ {- C}$$ (for any fixed $$C$$ Y $$n a infty$$).

I would appreciate any information related to sums of this form, similar sums or any understanding of how difficult this question may be.

Thank you!