[Paper Review] Lower bounds in some power sum problems

Johan Andersson|ArXiv.org|Apr 15, 2007

Analytic Number Theory Research5 references3 citations

TL;DR

This paper establishes improved lower bounds for the maximum of pure power sums over the unit circle using Fejér kernels and one-sided estimates. It proves that for $ j \geq 0 $, $ \min_{|z_k|=1} \max_{\nu=1,\dots,n^2+j} \left| \sum_{k=1}^n z_k^\nu \right| \geq \sqrt{ \frac{n + (1+j)(n-1)}{2(j + n^2)} } $, significantly improving prior estimates, especially for $ j \geq 2n^2 $. The method leverages harmonic analysis and non-negativity of Fejér kernels to derive tight bounds on exponential sums with unimodular roots.

ABSTRACT

We study the power sum problem max_{v=1,...,m} | sum_{k=1}^n z_k^v | and by using features of Fejer kernels we give new lower bounds in the case of unimodular complex numbers z_k and m cn^2 for constants c>1.

Motivation & Objective

• To improve existing lower bounds for the maximum of pure power sums $ \sum_{k=1}^n z_k^\nu $ with $ |z_k| = 1 $, particularly over intervals of length $ n^2 + j $.
• To establish tighter estimates for the minimal possible maximum of such sums using Fejér kernel techniques and one-sided $ L^2 $-type inequalities.
• To analyze the asymptotic behavior of the infimum of the maximum of power sums over intervals of length $ \lfloor \alpha n^2 \rfloor $, characterizing the growth rate via a function $ \Lambda(\alpha) $.

Proposed method

• Uses Fejér kernels $ F_{m+1}(x) = \sum_{\nu=-m}^m \left(1 - \frac{|\nu|}{m+1}\right) e(\nu x) $, which are non-negative and have known $ L^1 $-norm $ m+1 $, to control exponential sums.
• Applies one-sided estimates via decomposition $ g(\nu) = g^+(\nu) + g^-(\nu) $, where $ g^+ $ and $ g^- $ are the positive and negative parts of the sum.
• Derives lower bounds on $ \max g^+(\nu) $ using the identity $ \sum_{\nu=1}^m \left(1 - \frac{\nu}{m+1}\right) g^+(\nu) \geq \text{expression in } A, B, M $, where $ A = \sum b_k $, $ B = \sum b_k^2 $, and $ M $ bounds $ |g(\nu)| $.
• Applies these bounds to the pure power sum case by setting $ b_k = 1 $, so $ A = n $, $ B = n^2 - n $, and derives the key inequality involving $ j = m - n^2 $.
• Uses the identity $ |g(\nu)|^2 = n + h(\nu) $, where $ h(\nu) $ is a real-valued exponential sum over $ i \neq j $, to reduce the problem to bounding $ h(\nu) $.
• Employs a case analysis based on whether $ |h(\nu)| \leq B^2 $ or not, leading to two distinct lower bound expressions for $ \max |S(\nu)|^2 $.

Experimental results

Research questions

• RQ1 What is the best possible lower bound for $ \max_{\nu=1}^{n^2+j} \left| \sum_{k=1}^n z_k^\nu \right| $ when $ |z_k| = 1 $ and $ j \geq 0 $?
• RQ2 How do the bounds from Fejér kernel methods compare to known Turán-type estimates for power sums?
• RQ3 Can tighter bounds be derived for $ \max_{\nu=1}^{\lfloor \alpha n^2 \rfloor} \left| \sum z_k^\nu \right| $ as $ \alpha \to \infty $?
• RQ4 What is the optimal growth rate of the function $ \Lambda(\alpha) $ such that $ \inf_{|z_k|=1} \max_{\nu=1}^{\lfloor \alpha n^2 \rfloor} |S(\nu)| \geq \Lambda(\alpha) \sqrt{n} $?

Key findings

• For $ j \geq 0 $, the paper proves the lower bound $ \min_{|z_k|=1} \max_{\nu=1}^{n^2+j} \left| \sum z_k^\nu \right| \geq \sqrt{ \frac{n + (1+j)(n-1)}{2(j + n^2)} } $, which improves upon earlier estimates.
• For $ m = n^2 + j $, this bound is tighter than Turán's $ \sqrt{n} $ when $ j > 0 $, and for $ j = 0 $, it gives $ \geq \sqrt{ \frac{n+1}{2n} - \frac{1}{2n^2} } $, improving slightly on the known $ \sqrt{n} $.
• When $ j \geq 2n^2 $, the bound becomes $ \geq \sqrt{ \frac{2 - 2/\alpha - o(1)}{\sqrt{n}} } $, which improves further on the earlier expression.
• For $ \alpha \geq 3 $, the asymptotic lower bound is $ \left( \sqrt{2 - \frac{2}{\alpha}} - o(1) \right) \sqrt{n} $, showing that the optimal $ \Lambda(\alpha) $ satisfies $ \lim_{\alpha \to \infty} \Lambda(\alpha) = \sqrt{2} $.
• The paper shows that the true asymptotic for $ \max_{\nu=1}^{n^2} |S(\nu)| $ is $ \sim \sqrt{n} $, consistent with earlier results, but for $ \alpha > 1 $, the lower and upper bounds do not yet match.
• The lower bound in Corollary 3, $ \sqrt{ \frac{n+1}{2} - \frac{2n-1}{2(n^2+n-1)} } $, approximately halves the gap between known upper and lower bounds when $ n+1 $ is prime.