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Turán's method

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Number theory in mathematics
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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

s ν = ∑ n = 1 N b n z n ν   {\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 sν where | z n | ≥ 1 {\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 |sν| at least c(M, N)|s0| where

c ( M , N ) = ( ∑ k = 0 N − 1 ( M + k k ) 2 k ) − 1   . {\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 ( N 2 e ( M + N ) ) N − 1 {\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 sν where | z n | ≤ 1 {\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

| s ν | ≥ 2 ( N 8 e ( M + N ) ) N min 1 ≤ j ≤ N | ∑ n = 1 j b n |   . {\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

References

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