Turán's method

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.

Turán's first theorem

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.

Turán's second theorem

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

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

See also

• Turán's theorem in graph theory

References

• 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.