Turán's method

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In mathematics, Turán's method provides lower bounds for exponential sums and complex power sums. The method has been applied to problems in equidistribution.

The method applies to sums of the form

${\displaystyle s_{\nu }=\sum _{n=1}^{N}b_{n}z_{n}^{\nu }\ }$

where the b and z are complex numbers and ν runs over a range of integers. There are two main results, depending on the size of the complex numbers z.

The first result applies to sums s_ν where ${\displaystyle |z_{n}|\geq 1}$ for all n. For any range of ν of length N, say ν = M + 1, ..., M + N, there is some ν with |s_ν| at least c(M, N)|s₀| where

${\displaystyle c(M,N)=\left({\sum _{k=0}^{N-1}{\binom {M+k}{k}}2^{k}}\right)^{-1}\ .}$

The sum here may be replaced by the weaker but simpler ${\displaystyle \left({\frac {N}{2e(M+N)}}\right)^{N-1}}$.

We may deduce the Fabry gap theorem from this result.

The second result applies to sums s_ν where ${\displaystyle |z_{n}|\leq 1}$ for all n. Assume that the z are ordered in decreasing absolute value and scaled so that |z₁| = 1. Then there is some ν with

${\displaystyle |s_{\nu }|\geq 2\left({\frac {N}{8e(M+N)}}\right)^{N}\min _{1\leq j\leq N}\left\vert {\sum _{n=1}^{j}b_{n}}\right\vert \ .}$

• Turán's theorem in graph theory

• Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics. Vol. 84. Providence, RI: American Mathematical Society. ISBN 0-8218-0737-4. Zbl 0814.11001.