A real effective version of the Freĭman-Scourfield Theorem

by Tanja Küfner
Doctoral thesis
Date of Examination:2025-09-12
Date of issue:2025-11-04
Advisor:Prof. Dr. Jörg Brüdern
Referee:Prof. Dr. Jörg Brüdern
Referee:Prof. Dr. Damaris Schindler
Persistent Address: http://dx.doi.org/10.53846/goediss-11614

 

 

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Abstract

English

We study the solvability of a Diophantine equation with mixed real exponents for sufficiently large natural numbers. The connection to a condition on the sum over the reciprocals of the exponents was first shown by Freĭman and Scourfield and recently made effective by Brüdern and Wooley. We prove an effective statement for the case of real exponents, which involves flooring in the equation. To this end, we apply the theory of the Hardy-Littlewood circle method adapted to Diophantine inequalities. In the fourth chapter, we show a mean value estimate utilising diminishing ranges and prove a novel Weyl-type estimate for exponential sums with a non-integer exponent. The calculations on the arcs in the fifth chapter lead to conditions that are translated into the effective result in the last chapter.
Keywords: Analytic number theory; Diophantine equations; Diophantine inequalities; Hardy-Littlewood circle method; Freĭman-Scourfield theorem; Non-integer exponents
 
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