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 $$ Pr (| S_n | <e ^ {- 100n}) $$ 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!