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Interpolation of hypergeometric ratios in a global field of positive characteristic

Greg W. Anderson (2007)

Annales de l’institut Fourier

For each global field of positive characteristic we exhibit many examples of two-variable algebraic functions possessing properties consistent with a conjectural refinement of the Stark conjecture in the function field case recently proposed by the author. All the examples are Coleman units. We obtain our results by studying rank one shtukas in which both zero and pole are generic, i. e., shtukas not associated to any Drinfeld module.

On sums and products in a field

Guang-Liang Zhou, Zhi-Wei Sun (2022)

Czechoslovak Mathematical Journal

We study sums and products in a field. Let F be a field with ch ( F ) ≠ 2 , where ch ( F ) is the characteristic of F . For any integer k ≥ 4 , we show that any x ∈ F can be written as a 1 + ⋯ + a k with a 1 , ⋯ , a k ∈ F and a 1 ⋯ a k = 1 , and that for any α ∈ F ∖ { 0 } we can write every x ∈ F as a 1 ⋯ a k with a 1 , ⋯ , a k ∈ F and a 1 + ⋯ + a k = α . We also prove that for any x ∈ F and k ∈ { 2 , 3 , ⋯ } there are a 1 , ⋯ , a 2 k ∈ F such that a 1 + ⋯ + a 2 k = x = a 1 ⋯ a 2 k .

On the distribution of sparse sequences in prime fields and applications

Víctor Cuauhtemoc García (2013)

Journal de Théorie des Nombres de Bordeaux

In the present paper we investigate distributional properties of sparse sequences modulo almost all prime numbers. We obtain new results for a wide class of sparse sequences which in particular find applications on additive problems and the discrete Littlewood problem related to lower bound estimates of the L 1 -norm of trigonometric sums.

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