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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.

Contents

The method applies to sums of the form

s ν = ∑ n = 1 N b n z n ν  

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 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   .

The sum here may be replaced by the weaker but simpler ( N 2 e ( M + N ) ) 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 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 |   .

References

Turán's method Wikipedia
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