Tiling and weak tiling in $(\mathbb{Z}_p)^d$

D\'avid Matolcsi; G\'abor Somlai; Gergely Kiss; M\'at\'e Matolcsi

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Tiling and weak tiling in $(\mathbb{Z}_p)^d$

Gergely Kiss, Dávid Matolcsi, Máté Matolcsi, Gábor Somlai

Combinatorics · Submitted 11 Dec 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We characterize such 4-tuples for $d=1, 2$, and prove some partial results for $d=3$.

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arXiv ID2212.05513 (version 1)

Pith IDPZQG7FSU, Pith's permanent record of this paper and its review

CitationD\'avid Matolcsi, G\'abor Somlai, Gergely Kiss, M\'at\'e Matolcsi. "Tiling and weak tiling in $(\mathbb{Z}_p)^d$." arXiv:2212.05513 (2022). https://pith.science/paper/PZQG7FSU

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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,979 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.71 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.15 · mentions code or data availability 0.01.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary $p$-groups $(\mathbb{Z}_p)^d$, we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for $d=1, 2$, and prove some partial results for $d=3$.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups.	background 0.81	0.66	0.07	asserted 0.51
s1	In particular, in elementary $p$-groups $(\mathbb{Z}_p)^d$, we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets.	contribution 0.96	0.94	0.16	asserted 1.00
s2	We characterize such 4-tuples for $d=1, 2$, and prove some partial results for $d=3$.	result 0.56	0.96	0.37	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 1.00.

What was examined, and what was not

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arXiv listing abstract v1

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full text
figures and tables
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P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary $p$-groups $(\mathbb{Z}_p)^d$, we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for $d=1, 2$, and prove some partial results for $d=3$.

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1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it

Cites it as background 1 paper

This method involves the notion of a positive definite function, which is not new in this conjecture [ref] .

A linear programming approach to Fuglede's conjecture in $\mathbb{Z}_p^3$ · Introduction · 0.83

What it cites

Cites as background 2 papers

For finite abelian groups, several results have been discovered in recent years [ref] .

Fuglede's conjecture holds in $\mathbb{Z}_{p}\times\mathbb{Z}_{p^{n}}$ · Introduction · 0.98
For finite abelian groups, several results have been discovered in recent years [ref] .

Spectral sets and weak tiling · Introduction · 0.98

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