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

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[edit]

The first result applies to sums sν where 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(MN)|s0| where

The sum here may be replaced by the weaker but simpler .

We may deduce the Fabry gap theorem from this result.

Turán's second theorem[edit]

The second result applies to sums sν where 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

See also[edit]