# On some power sum problems of Montgomery and Turan

@article{Andersson2007OnSP, title={On some power sum problems of Montgomery and Turan}, author={John Andersson}, journal={arXiv: Number Theory}, year={2007} }

We use an estimate for character sums over finite fields of Katz to solve open problems of Montgomery and Turan. Let h=>2 be an integer. We prove that inf_{|z_k| => 1} max_{v=1,...,n^h} |sum_{k=1}^n z_k^v| <= (h-1+o(1)) sqrt n. This gives the right order of magnitude for the quantity and improves on a bound of Erdos-Renyi by a factor of the order sqrt log n.

## 3 Citations

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- Mathematics, Computer Science
- 2006

In this paper we prove that inf_{|z_k| => 1} max_{v=1,...,n^2} |sum_{k=1}^n z_k^v| = sqrt n+O(n^{0.2625+epsilon}). This improves on the bound O(sqrt (n log n)) of Erdos and Renyi. In the special case…

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We solve an analog of Chui’s conjecture on the simplest fractions (i.e., sums of Cauchy kernels with unit coefficients) in weighted (Hilbert) Bergman spaces. Namely, for a wide class of weights, we…

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A new explicit construction of n×N matrices satisfying the Restricted Isometry Property is given, which overcomes the natural barrier k = O(n1/2) for proofs based on small coherence, which is used in all previous explicit constructions of RIP matrices.

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