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

 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[edit]

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

 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 \left({ \frac{N}{2e(M+N)} }\right)^{N-1}.

We may deduce Fabry's gap theorem from this result.

Turán's second theorem[edit]

The second result applies to sums sν where |z_n| \le 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_\nu| \ge 2 \left({ \frac{N}{8e(M+N)} }\right)^N \min_{1\le j\le N} \left\vert{\sum_{n=1}^j b_n }\right\vert \ .

See also[edit]

References[edit]