A Halász-type theorem for permutation anticoncentration
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918279798325248 |
|---|---|
| author | Hunter, Zach Pohoata, Cosmin Zhu, Daniel G. |
| author_facet | Hunter, Zach Pohoata, Cosmin Zhu, Daniel G. |
| contents | Given a set $A=\{a_1,\ldots,a_n\}$ of real numbers and real coefficients $b_1,\ldots,b_n$, consider the distribution of the sum obtained by pairing the $a_i$'s with the $b_i$'s according to a uniformly random permutation. A recent theorem of Pawlowski shows that as soon as the coefficients are not all equal, this distribution is always spread out at scale $n^{-1}$: no single value can occur with probability larger than $\frac{1}{2\lceil n/2\rceil + 1}$, and this bound is sharp in general.
We show that stronger anticoncentration holds when the coefficients have additional diversity. We quantify the structure of the coefficient multiset by a simple statistic depending on its multiplicity profile, and prove that the maximum point mass of the permuted sum decays polynomially faster as this statistic grows. In particular, when the coefficients are all distinct we obtain a bound of $n^{-5/2+o(1)}$, which can be regarded as an analogue of a classical theorem of Erdős and Moser. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06019 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Halász-type theorem for permutation anticoncentration Hunter, Zach Pohoata, Cosmin Zhu, Daniel G. Combinatorics Probability 05D40, 60C05 Given a set $A=\{a_1,\ldots,a_n\}$ of real numbers and real coefficients $b_1,\ldots,b_n$, consider the distribution of the sum obtained by pairing the $a_i$'s with the $b_i$'s according to a uniformly random permutation. A recent theorem of Pawlowski shows that as soon as the coefficients are not all equal, this distribution is always spread out at scale $n^{-1}$: no single value can occur with probability larger than $\frac{1}{2\lceil n/2\rceil + 1}$, and this bound is sharp in general. We show that stronger anticoncentration holds when the coefficients have additional diversity. We quantify the structure of the coefficient multiset by a simple statistic depending on its multiplicity profile, and prove that the maximum point mass of the permuted sum decays polynomially faster as this statistic grows. In particular, when the coefficients are all distinct we obtain a bound of $n^{-5/2+o(1)}$, which can be regarded as an analogue of a classical theorem of Erdős and Moser. |
| title | A Halász-type theorem for permutation anticoncentration |
| topic | Combinatorics Probability 05D40, 60C05 |
| url | https://arxiv.org/abs/2601.06019 |