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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.03625 |
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| _version_ | 1866914705321230336 |
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| author | Mohan Mistri, Raj Kumar Pandey, Ram Krishna |
| author_facet | Mohan Mistri, Raj Kumar Pandey, Ram Krishna |
| contents | Given a positive integer $h$ and a nonempty finite set of integers $A=\{a_{1},a_{2},\ldots,a_{k}\}$, the restricted $h$-fold signed sumset of $A$, denoted by $h^{\wedge}_{\pm}A$, is defined as $$h^{\wedge}_{\pm}A=\left\lbrace \sum_{i=1}^{k} λ_{i} a_{i}: λ_{i} \in \left\lbrace -1, 0, 1\right\rbrace \ \text{for} \ i= 1, 2, \ldots, k \ \text{and} \ \sum_{i=1}^{k} \left| λ_{i} \right| =h\right\rbrace.$$ The direct problem associated with this sumset is to find the optimal lower bound of $|h^{\wedge}_{\pm}A|$, and the inverse problem associated with this sumset is to determine the structure of the underlying set $A$, when $|h^{\wedge}_{\pm}A|$ attains the optimal lower bound. Bhanja, Komatsu and Pandey studied the direct and inverse problem for the restricted $h$-fold signed sumset for $h=2, 3$, and $k$ and conjectured some direct and inverse results for $h \geq 4$. In this paper, we prove these conjectures for $h=4$. We also prove the direct and inverse theorems for arbitrary $h$ under certain restrictions on the set $A$ which are particular cases of the conjectures. Moreover, we prove these conjectures for arithmetic progressions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03625 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some direct and inverse problems for the Restricted Signed sumset in set of integers Mohan Mistri, Raj Kumar Pandey, Ram Krishna Number Theory Given a positive integer $h$ and a nonempty finite set of integers $A=\{a_{1},a_{2},\ldots,a_{k}\}$, the restricted $h$-fold signed sumset of $A$, denoted by $h^{\wedge}_{\pm}A$, is defined as $$h^{\wedge}_{\pm}A=\left\lbrace \sum_{i=1}^{k} λ_{i} a_{i}: λ_{i} \in \left\lbrace -1, 0, 1\right\rbrace \ \text{for} \ i= 1, 2, \ldots, k \ \text{and} \ \sum_{i=1}^{k} \left| λ_{i} \right| =h\right\rbrace.$$ The direct problem associated with this sumset is to find the optimal lower bound of $|h^{\wedge}_{\pm}A|$, and the inverse problem associated with this sumset is to determine the structure of the underlying set $A$, when $|h^{\wedge}_{\pm}A|$ attains the optimal lower bound. Bhanja, Komatsu and Pandey studied the direct and inverse problem for the restricted $h$-fold signed sumset for $h=2, 3$, and $k$ and conjectured some direct and inverse results for $h \geq 4$. In this paper, we prove these conjectures for $h=4$. We also prove the direct and inverse theorems for arbitrary $h$ under certain restrictions on the set $A$ which are particular cases of the conjectures. Moreover, we prove these conjectures for arithmetic progressions. |
| title | Some direct and inverse problems for the Restricted Signed sumset in set of integers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2403.03625 |