Distinct permutation dot products
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917209432915968 |
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| author | Pohoata, Cosmin |
| author_facet | Pohoata, Cosmin |
| contents | We show that for any two sets of reals numbers $A=\{a_1,\dots,a_n\}$ and $B=\{b_1,\dots,b_n\}$, the sums of the form $\sum_{i=1}^n a_i\,b_{π(i)}$ always take on $Ω(n^{3})$ distinct values, as we range over all permutations $π\in S_n$.
An important ingredient is a ``supportive'' version of Halász's anticoncentration theorem from Littlewood-Offord theory, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12445 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Distinct permutation dot products Pohoata, Cosmin Combinatorics We show that for any two sets of reals numbers $A=\{a_1,\dots,a_n\}$ and $B=\{b_1,\dots,b_n\}$, the sums of the form $\sum_{i=1}^n a_i\,b_{π(i)}$ always take on $Ω(n^{3})$ distinct values, as we range over all permutations $π\in S_n$. An important ingredient is a ``supportive'' version of Halász's anticoncentration theorem from Littlewood-Offord theory, which may be of independent interest. |
| title | Distinct permutation dot products |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.12445 |