Published March 2023
| public
Journal Article
An analogue of Furstenberg–Sárközy's theorem and an alternative solution to Waring's problem over finite fields
- Creators
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Demiroğlu Karabulut, Yeşi̇m
Chicago
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Abstract
In this paper, we use Cayley digraphs to obtain some new self-contained proofs for Waring's problem over finite fields, proving that any element of a finite field F_q can be written as a sum of m many kth powers as long as q > k^(2m/m-1); and we also compute the smallest positive integers such that every element of can be written as a sum of many powers for all too small to be covered by the above mentioned results when 2 ⩽ k ⩽ 37. In the process of developing the proofs mentioned above, we arrive at another result (providing a finite field analogue of Furstenberg–Sárközy's Theorem) showing that any subset of a finite field F_q for which |E| > qk/√q̅-̅1̅ must contain at least two distinct elements whose difference is a power.
Additional Information
© 2022 Elsevier. I would like to thank Jonathan Pakianathan and David Covert for suggesting this problem. I also would like to thank Stephan Ramon Garcia for his aid to prove Lemma A.1. Many thanks to the University of Rochester CIRC team for letting me use their supercomputer. Last but not least, I am also indebted to the anonymous referee for many useful suggestions. Data availability. I shared the computer code in the appendix B of the text. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Additional details
- Eprint ID
- 120659
- Resolver ID
- CaltechAUTHORS:20230404-414969100.3
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2023-04-11Created from EPrint's datestamp field
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2023-04-11Created from EPrint's last_modified field