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Equidistribution of higher dimensional generalized Dedekind sums and exponential sums. (English) Zbl 1465.11099

Summary: We consider generalized Dedekind sums in dimension \(n\), defined as sum of products of values of periodic Bernoulli functions. For the generalized Dedekind sums, we associate a Laurent polynomial. Using this, we associate an exponential sum of a Laurent polynomial to the generalized Dedekind sums and show that this exponential sum has a nontrivial bound that is sufficient to fulfill the equidistribution criterion of Weyl and thus the fractional part of the generalized Dedekind sums are equidistributed in \(\mathbb{R}/\mathbb{Z}\).

MSC:

11F20 Dedekind eta function, Dedekind sums
11L03 Trigonometric and exponential sums (general theory)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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