Explicit Constructions of Maximal 3-Zero-Sum-Free Subsets in $ (\mathbb{Z}/4\mathbb{Z})^n $
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909767965868032 |
|---|---|
| author | Vera, Alfonso Davila |
| author_facet | Vera, Alfonso Davila |
| contents | We address a problem posed by Nathan Kaplan in the 2014 Combinatorial and Additive Number Theory session: finding the largest subset $H \subseteq (\mathbb{Z}/4\mathbb{Z})^n$ with no distinct $x, y, z \in H$ such that $x + y + z \equiv 0 \pmod{4}$. For even-order abelian groups, a standard $|G|/2$ lower bound applies. We prove this is optimal for $G = (\mathbb{Z}/4\mathbb{Z})^n$ using a pair-counting argument, with an explicit construction of vectors with first coordinate odd (1 or 3 mod 4), yielding size $2 \times 4^{n-1} = 4^n / 2$ and density 0.5, verified for $n \leq 10$. An AI-assisted hybrid greedy-genetic algorithm rediscovers this optimal size, highlighting its potential in combinatorial search. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01735 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit Constructions of Maximal 3-Zero-Sum-Free Subsets in $ (\mathbb{Z}/4\mathbb{Z})^n $ Vera, Alfonso Davila Combinatorics Number Theory 05D99, 11B75, 20K01 G.2.1; I.2.8 We address a problem posed by Nathan Kaplan in the 2014 Combinatorial and Additive Number Theory session: finding the largest subset $H \subseteq (\mathbb{Z}/4\mathbb{Z})^n$ with no distinct $x, y, z \in H$ such that $x + y + z \equiv 0 \pmod{4}$. For even-order abelian groups, a standard $|G|/2$ lower bound applies. We prove this is optimal for $G = (\mathbb{Z}/4\mathbb{Z})^n$ using a pair-counting argument, with an explicit construction of vectors with first coordinate odd (1 or 3 mod 4), yielding size $2 \times 4^{n-1} = 4^n / 2$ and density 0.5, verified for $n \leq 10$. An AI-assisted hybrid greedy-genetic algorithm rediscovers this optimal size, highlighting its potential in combinatorial search. |
| title | Explicit Constructions of Maximal 3-Zero-Sum-Free Subsets in $ (\mathbb{Z}/4\mathbb{Z})^n $ |
| topic | Combinatorics Number Theory 05D99, 11B75, 20K01 G.2.1; I.2.8 |
| url | https://arxiv.org/abs/2509.01735 |