Near-optimal density theorems for large dilates of large point configurations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910149275287552 |
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| author | Kovač, Vjekoslav Sepčić, Adian Anibal Santos |
| author_facet | Kovač, Vjekoslav Sepčić, Adian Anibal Santos |
| contents | We study density thresholds that force a measurable set $E\subseteq\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration. We prove a lower bound of the form $1-O((\log n)/n)$, which matches the known upper bound up to the logarithmic factor, thus essentially resolving a problem posed by Falconer, Yavicoli, and the first author of the present paper. We also study the same problem for embeddings of $n$-point configurations into $\mathbb{R}^d$ equipped with the $\ell^p$ norm, obtaining an asymptotically sharp bound $1-1/n+o(1/n)$, as soon as $p\in(1,\infty)\setminus\{2\}$. In the proof of the former estimate we use equidistribution of polynomial sequences modulo $1$ combined with probabilistic thinning. The proof of the latter estimate relies on the geometry of the $\ell^p$ spaces for $p\neq2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_18544 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Near-optimal density theorems for large dilates of large point configurations Kovač, Vjekoslav Sepčić, Adian Anibal Santos Classical Analysis and ODEs Combinatorics Number Theory We study density thresholds that force a measurable set $E\subseteq\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration. We prove a lower bound of the form $1-O((\log n)/n)$, which matches the known upper bound up to the logarithmic factor, thus essentially resolving a problem posed by Falconer, Yavicoli, and the first author of the present paper. We also study the same problem for embeddings of $n$-point configurations into $\mathbb{R}^d$ equipped with the $\ell^p$ norm, obtaining an asymptotically sharp bound $1-1/n+o(1/n)$, as soon as $p\in(1,\infty)\setminus\{2\}$. In the proof of the former estimate we use equidistribution of polynomial sequences modulo $1$ combined with probabilistic thinning. The proof of the latter estimate relies on the geometry of the $\ell^p$ spaces for $p\neq2$. |
| title | Near-optimal density theorems for large dilates of large point configurations |
| topic | Classical Analysis and ODEs Combinatorics Number Theory |
| url | https://arxiv.org/abs/2604.18544 |