Quasipolynomial density bounds for K-point configurations in ℤ^d

Abstract

Let d,K,N∈ ℕ with K≥ 3 and d≥ 4K+4. Let Δ⊂ ℤ^d be the vertex set of a nondegenerate (K−1)-simplex, and let A⊆[N]^d contain no nontrivial similar copy of Δ. We prove that |A|≪_Δ,d N^dexp (−c_Δ,d√(log N)) improving upon a polylogarithmic bound due to Magyar. We perform a density increment argument using the circle method, and we introduce a ``cut operator'' method to decouple the weighted exponential sum over the system of quadratic forms describing the simplex. Our proof combines ideas from graph theory, functional analysis, and the geometry of numbers. In the process, we apply Finner's fractional form of Hölder's inequality, the analytic large sieve, and Kim's mean value formula for primitive lattice flags.

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BibTeX RIS

BibTeX

@misc{lott2026quasipolynomial,
  title = {{Quasipolynomial density bounds for $K$-point configurations in $\mathbbZ^d$}},
  author = {Andrew Lott and Ákos Magyar and Nagendar Reddy Ponagandla},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13126},
  eprint = {2609.13126},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13126},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Quasipolynomial density bounds for K-point configurations in ℤ^d
AU  - Andrew Lott
AU  - Ákos Magyar
AU  - Nagendar Reddy Ponagandla
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.13126
UR  - https://arxiv.org/abs/2609.13126
AB  - Let d,K,N∈ ℕ with K≥ 3 and d≥ 4K+4. Let Δ⊂ ℤ^d be the vertex set of a nondegenerate (K−1)-simplex, and let A⊆[N]^d contain no nontrivial similar copy of Δ. We prove that |A|≪_Δ,d N^dexp (−c_Δ,d√(log N)) improving upon a polylogarithmic bound due to Magyar. We perform a density increment argument using the circle method, and we introduce a ``cut operator'' method to decouple the weighted exponential sum over the system of quadratic forms describing the simplex. Our proof combines ideas from graph theory, functional analysis, and the geometry of numbers. In the process, we apply Finner's fractional form of Hölder's inequality, the analytic large sieve, and Kim's mean value formula for primitive lattice flags.
ER  -

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