The fourth positive element in the greedy $B_h$-set
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929514068574208 |
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| author | Nathanson, Melvyn B. O'Bryant, Kevin |
| author_facet | Nathanson, Melvyn B. O'Bryant, Kevin |
| contents | For $h \geq 1$, a $B_h$-set is a set of integers such that every integer $n$ has at most one representation in the form $n = a_{i_1} + \cdots + a_{i_h}$, where $a_{i_r} \in A$ for all $r = 1,\ldots, h$ and $a_{i_1} \leq \ldots \leq a_{i_h}$. The greedy $B_h$-set is the infinite set of nonnegative integers $\{a_0(h), a_1(h), a_2(h), \ldots \}$ constructed as follows: If $a_0(h) = 0$ and $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h) \}$ is a $B_h$-set, then $a_{k+1}(h)$ is the least positive integer such that $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h), a_{k+1}(h) \}$ is a $B_h$-set. Then $a_1(h) = 1$, $a_2(h) = h+1$, and $a_3(h) = h^2+h+1$ for all $h$. This paper proves that $a_4(h)$, the fourth term of the greedy $B_h$-set is $\left( h^3 + 3h^2 + 3h + 1\right) /2$ if $h$ is odd and $\left( h^3 + 2h^2 + 3h + 2\right) /2$ if $h$ is even. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_14021 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The fourth positive element in the greedy $B_h$-set Nathanson, Melvyn B. O'Bryant, Kevin Number Theory Combinatorics 11B13, 11B34, 11B75, 11P99 For $h \geq 1$, a $B_h$-set is a set of integers such that every integer $n$ has at most one representation in the form $n = a_{i_1} + \cdots + a_{i_h}$, where $a_{i_r} \in A$ for all $r = 1,\ldots, h$ and $a_{i_1} \leq \ldots \leq a_{i_h}$. The greedy $B_h$-set is the infinite set of nonnegative integers $\{a_0(h), a_1(h), a_2(h), \ldots \}$ constructed as follows: If $a_0(h) = 0$ and $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h) \}$ is a $B_h$-set, then $a_{k+1}(h)$ is the least positive integer such that $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h), a_{k+1}(h) \}$ is a $B_h$-set. Then $a_1(h) = 1$, $a_2(h) = h+1$, and $a_3(h) = h^2+h+1$ for all $h$. This paper proves that $a_4(h)$, the fourth term of the greedy $B_h$-set is $\left( h^3 + 3h^2 + 3h + 1\right) /2$ if $h$ is odd and $\left( h^3 + 2h^2 + 3h + 2\right) /2$ if $h$ is even. |
| title | The fourth positive element in the greedy $B_h$-set |
| topic | Number Theory Combinatorics 11B13, 11B34, 11B75, 11P99 |
| url | https://arxiv.org/abs/2311.14021 |