Sarkozy's theorem in function fields
<p>Sárközy proved that dense sets of integers contain two elements differing by a kth power. The bounds in quantitative versions of this theorem are rather weak compared to what is expected. We prove a version of Sárközy’s theorem for polynomials over Fq with polynomial dependencies in the par...
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| Format: | Journal article |
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Oxford University Press
2016
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| _version_ | 1826273900832489472 |
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| author | Green, B |
| author_facet | Green, B |
| author_sort | Green, B |
| collection | OXFORD |
| description | <p>Sárközy proved that dense sets of integers contain two elements differing by a kth power. The bounds in quantitative versions of this theorem are rather weak compared to what is expected. We prove a version of Sárközy’s theorem for polynomials over Fq with polynomial dependencies in the parameters</p> <br/> <p>More precisely, let Pq,n be the space of polynomials over Fq of degree < n in an indeterminate T. Let k ⩾ 2 be an integer and let q be a prime power. Set c(k, q) := (2k^2 Dq (k)^2 log q)^−1, where Dq (k) is the sum of the digits of k in base q. If A ⊂ Pq,n is a set with |A| > 2q^(1−c(k,q))n, then A contains distinct polynomials p(T ), p′(T ) such that p(T ) − p′(T ) = b(T)^k for some b ∈ Fq [T ].</p> |
| first_indexed | 2024-03-06T22:35:16Z |
| format | Journal article |
| id | oxford-uuid:59adef99-1289-4137-bccd-5ed1f643441b |
| institution | University of Oxford |
| last_indexed | 2024-03-06T22:35:16Z |
| publishDate | 2016 |
| publisher | Oxford University Press |
| record_format | dspace |
| spelling | oxford-uuid:59adef99-1289-4137-bccd-5ed1f643441b2022-03-26T17:11:09ZSarkozy's theorem in function fieldsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:59adef99-1289-4137-bccd-5ed1f643441bSymplectic Elements at OxfordOxford University Press2016Green, B<p>Sárközy proved that dense sets of integers contain two elements differing by a kth power. The bounds in quantitative versions of this theorem are rather weak compared to what is expected. We prove a version of Sárközy’s theorem for polynomials over Fq with polynomial dependencies in the parameters</p> <br/> <p>More precisely, let Pq,n be the space of polynomials over Fq of degree < n in an indeterminate T. Let k ⩾ 2 be an integer and let q be a prime power. Set c(k, q) := (2k^2 Dq (k)^2 log q)^−1, where Dq (k) is the sum of the digits of k in base q. If A ⊂ Pq,n is a set with |A| > 2q^(1−c(k,q))n, then A contains distinct polynomials p(T ), p′(T ) such that p(T ) − p′(T ) = b(T)^k for some b ∈ Fq [T ].</p> |
| spellingShingle | Green, B Sarkozy's theorem in function fields |
| title | Sarkozy's theorem in function fields |
| title_full | Sarkozy's theorem in function fields |
| title_fullStr | Sarkozy's theorem in function fields |
| title_full_unstemmed | Sarkozy's theorem in function fields |
| title_short | Sarkozy's theorem in function fields |
| title_sort | sarkozy s theorem in function fields |
| work_keys_str_mv | AT greenb sarkozystheoreminfunctionfields |