Additive triples in groups of odd prime order
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911870546345984 |
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| author | Huczynska, Sophie Jedwab, Jonathan Johnson, Laura |
| author_facet | Huczynska, Sophie Jedwab, Jonathan Johnson, Laura |
| contents | Let $p$ be an odd prime. For nontrivial proper subsets $A,B$ of $\mathbb{Z}_p$ of cardinality $s,t$, respectively, we count the number $r(A,B,B)$ of additive triples, namely elements of the form $(a, b, a+b)$ in $A \times B \times B$. For given $s,t$, what is the spectrum of possible values for $r(A,B,B)$? In the special case $A=B$, the additive triple is called a Schur triple. Various authors have given bounds on the number $r(A,A,A)$ of Schur triples, and shown that the lower and upper bound can each be attained by a set $A$ that is an interval of $s$ consecutive elements of $\mathbb{Z}_p$. However, there are values of $p,s$ for which not every value between the lower and upper bounds is attainable. We consider here the general case where $A,B$ can be distinct. We use Pollard's generalization of the Cauchy-Davenport Theorem to derive bounds on the number $r(A,B,B)$ of additive triples. In contrast to the case $A=B$, we show that every value of $r(A,B,B)$ from the lower bound to the upper bound is attainable: each such value can be attained when $B$ is an interval of $t$ consecutive elements of $\mathbb{Z}_p$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_04638 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Additive triples in groups of odd prime order Huczynska, Sophie Jedwab, Jonathan Johnson, Laura Combinatorics 05A15, 05E15 Let $p$ be an odd prime. For nontrivial proper subsets $A,B$ of $\mathbb{Z}_p$ of cardinality $s,t$, respectively, we count the number $r(A,B,B)$ of additive triples, namely elements of the form $(a, b, a+b)$ in $A \times B \times B$. For given $s,t$, what is the spectrum of possible values for $r(A,B,B)$? In the special case $A=B$, the additive triple is called a Schur triple. Various authors have given bounds on the number $r(A,A,A)$ of Schur triples, and shown that the lower and upper bound can each be attained by a set $A$ that is an interval of $s$ consecutive elements of $\mathbb{Z}_p$. However, there are values of $p,s$ for which not every value between the lower and upper bounds is attainable. We consider here the general case where $A,B$ can be distinct. We use Pollard's generalization of the Cauchy-Davenport Theorem to derive bounds on the number $r(A,B,B)$ of additive triples. In contrast to the case $A=B$, we show that every value of $r(A,B,B)$ from the lower bound to the upper bound is attainable: each such value can be attained when $B$ is an interval of $t$ consecutive elements of $\mathbb{Z}_p$. |
| title | Additive triples in groups of odd prime order |
| topic | Combinatorics 05A15, 05E15 |
| url | https://arxiv.org/abs/2405.04638 |