컨텐츠 시작
학술대회/행사
초록검색
| 제출번호(No.) | 0160 |
|---|---|
| 분류(Section) | Contributed Talk |
| 분과(Session) | (DM) Discrete Mathematics (DM) |
| 발표시간(Time) | 19th-B-13:20 -- 13:40 |
| 영문제목 (Title(Eng.)) |
Improved upper bounds for the largest size of Diophantine m-tuples |
| 저자(Author(s)) |
Seoyoung Kim1, Chi Hoi Yip2, Semin Yoo3 University of Göttingen1, University of British Columbia2, IBS-DIMAG3 |
| 초록본문(Abstract) | A set $\{a_{1},a_{2},\ldots , a_{m}\}$ of distinct positive integers is a \textit{Diophantine $m$-tuple} with property $D_{k}(n)$ if the product of any two distinct elements in the set is $n$ less than a $k$-th power. One may wonder what is the largest size, $M_k(n)$, of such a tuple. In this talk, we provide a substantial improvement on a recent result by Dixit, Kim, and Murty (2022) on the upper bound of $M_k(n)$. In particular, we show $M_k(n)=o(\log n)$ for a specially chosen sequence of $k$ and $n$ tending to infinity, breaking the $\log n$ barrier unconditionally. Our proof is a combination of Stepanov's method and Gallagher's larger sieve. This is joint work with Seoyoung Kim, and Chi Hoi Yip. |
| 분류기호 (MSC number(s)) |
Primary 11B30, 11D72; Secondary 11D45, 11N36, 11L40 |
| 키워드(Keyword(s)) | Diophantine tuples, shifted multiplicative subgroup |
| 강연 형태 (Language of Session (Talk)) |
English |