Thank you so much.

It is a quite tight bounds, great! Actually, my final goal is to get the bounds when $s_n$ is a cross-correlation of a ML-sequences and a reciprocal version of it, which is a reversed ML-sequences. In this case, $s_n$ is not a ML-sequences, but there are some papers derived the bounds of cross-correlation of those two, it's just not a exponential sums. 'Cross-Correlations of Reverse Maximal-length Shift Register Sequences', J. A. Dowlingl and R. McHiece, but would it be possible to get a bounds of it?

@GerryMyerson Thanks for letting me to know that.

en.wikipedia.org/wiki/Maximum_length_sequence basically, there are 3 properties : run, balance, correlation. $\alpha$ holds those properties. Sorry for the confusion.

Right, that would be the upper bound, but I forgot to mention that alpha follows pseudo random properties. My apologize...