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On minima of the absolute value of certain random exponential sums
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 122, Number 3, June 2000
- pp. 483-514
- 10.1353/ajm.2000.0022
- Article
- Additional Information
Let Tn(x) = Σnj=1 ±e2πij2x where ± stands for a random choice of sign with equal probability. It is shown here that with high probability minxε[0,1) |Tn(x)| < n-σ provided n is large and σ < 1/12. Similar results are proved for other powers than squares. The problem of determining the optimal σ is open. For the case Tn(x)= Σnj=1rje2πijdx, where d = 2, 3,... is fixed and with standard normal rj we show that the minima are typically on the order of n-d+½ with high probability and for large n.