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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2312.10910 |
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| _version_ | 1866929331128762368 |
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| author | O'Bryant, Kevin |
| author_facet | O'Bryant, Kevin |
| contents | A set $A$ of nonnegative integers is called a $B_h$-set if every solution to $a_1+\dots+a_h = b_1+\dots+b_h$, where $a_i,b_i \in A$, has $\{a_1,\dots,a_h\}=\{b_1,\dots,b_h\}$ (as multisets). Let $γ_k(h)$ be the $k$-th positive element of the greedy $B_h$-set. We give a nontrivial lower bound on $γ_5(h)$, and a nontrivial upper bound on $γ_k(h)$ for $k\ge 5$. Specifically, $\frac 18 h^4 +\frac12 h^3 \le γ_5(h) \le 0.467214 h^4+O(h^3)$, although we conjecture that $γ_5(h)=\frac13 h^4 +O(h^3)$. We show that $γ_k(h) \ge \frac{1}{k!} h^{k-1} + O(h^{k-2})$ for $k\ge 1$ and $γ_k(h) \le α_k h^{k-1}+O(h^{k-2})$, where $α_6 := 0.382978$, $α_7 := 0.269877$, and for $k\ge 7$, $α_{k+1} := \frac{1}{2^k k!} \sum_{j=0}^{k-1} \binom{k-1}j\binom kj 2^j$. This work begins with a thorough introduction and concludes with a section of open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10910 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bounds for Greedy $B_h$-sets O'Bryant, Kevin Number Theory Combinatorics 11B13 A set $A$ of nonnegative integers is called a $B_h$-set if every solution to $a_1+\dots+a_h = b_1+\dots+b_h$, where $a_i,b_i \in A$, has $\{a_1,\dots,a_h\}=\{b_1,\dots,b_h\}$ (as multisets). Let $γ_k(h)$ be the $k$-th positive element of the greedy $B_h$-set. We give a nontrivial lower bound on $γ_5(h)$, and a nontrivial upper bound on $γ_k(h)$ for $k\ge 5$. Specifically, $\frac 18 h^4 +\frac12 h^3 \le γ_5(h) \le 0.467214 h^4+O(h^3)$, although we conjecture that $γ_5(h)=\frac13 h^4 +O(h^3)$. We show that $γ_k(h) \ge \frac{1}{k!} h^{k-1} + O(h^{k-2})$ for $k\ge 1$ and $γ_k(h) \le α_k h^{k-1}+O(h^{k-2})$, where $α_6 := 0.382978$, $α_7 := 0.269877$, and for $k\ge 7$, $α_{k+1} := \frac{1}{2^k k!} \sum_{j=0}^{k-1} \binom{k-1}j\binom kj 2^j$. This work begins with a thorough introduction and concludes with a section of open problems. |
| title | Bounds for Greedy $B_h$-sets |
| topic | Number Theory Combinatorics 11B13 |
| url | https://arxiv.org/abs/2312.10910 |