Terence Tao
We give an example of a set $Ω\subset \R^5$ which is a finite union of unit cubes, such that $L^2(Ω)$ admits an orthonormal basis of exponentials $\{\frac{1}{|Ω|^{1/2}} e^{2πi ξ_j \cdot x}: ξ_j \in Λ\}$ for some discrete set $Λ\subset \R^5$, but which does not tile $\R^5$ by translations. This answers a conjecture of Fuglede in the negative, at least in 5 and higher dimensions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Fuglede's conjecture states that a subset $Ω\subseteq\mathbb{R}^{n}$ of positive and finite Lebesgue measure is a spectral set if and only if it tiles $\mathbb{R}^{n}$ by translation. The conjecture does not hold in both directions for $\mathbb{R}^n$, $n\ge3$. However, this conjecture remains open in $\mathbb{R}$ and $\mathbb{R}^2$. Cyclic groups play important roles in the study of Fuglede's conjecture in $\mathbb{R}$. In this paper, we introduce a new tool to study the spectral sets in cyclic groups. In particular, we prove that Fuglede's conjecture holds in $\mathbb{Z}_{p^{n}qr}$.
Fuglede's conjecture states that for a subset $Ω$ of a locally compact abelian group $G$ with positive and finite Haar measure, there exists a subset of the dual group of $G$ which is an orthogonal basis of $L^{2}(Ω)$ if and only if it tiles the group by translation. In this paper, we prove a divisibility property for a set in $\mathbb{Z}_{p}\times\mathbb{Z}_{p^{n}}$. Then using the divisibility property and equi-distributed property, we prove that Fuglede's conjecture holds in the group $\mathbb{Z}_{p}\times\mathbb{Z}_{p^{n}}$.
For p=5,7, we show that a subset \(E \subset \F_p^3\) is spectral if and only if E tiles \(\F_p^3\) by translation. Additionally, we give an alternate proof that the conjecture holds for p=3.
In this note we give an example of a set $\W\subset \R^4$ such that $L^2(\W)$ admits an orthonormal basis of exponentials $\{\frac{1}{|\W |^{1/2}}e^{2πi x, ξ}\}_{ξ\inŁ}$ for some set $Ł\subset\R^4$, but which does not tile $\R^4$ by translations. This improves Tao's recent 5-dimensional example, and shows that one direction of Fuglede's conjecture fails already in dimension 4. Some common properties of translational tiles and spectral sets are also proved.
We describe computer programs accompanying our paper and show that running them suffices to verify Fuglede's conjecture in $\mathbb{Z}_2^5$ and $\mathbb{Z}_2^6$.
Both directions of Fuglede's conjecture fail in dimension two
研究富格莱德猜想在二维空间的情况，通过在二阶有限阿贝尔群中构造特定子集，利用有限到无限转移原理得到\(\R^2\)中有界子集，证明该猜想在二维空间中两个方向都不成立。
The Fuglede conjecture states that a set is spectral if and only if it tiles by translation. The conjecture was disproved by T. Tao for dimensions 5 and higher by giving a counterexample in $\mathbb{Z}_3^5$. We present a computer program that determines that the Fuglede conjecture holds in $\mathbb{Z}_5^3$ by exhausting the search space. A. Iosevich, A. Mayeli and J. Pakianathan showed that the Fuglede conjecture holds over prime fields when the dimension does not exceed 2. The question for dimension 3 was previously addressed by Aten et al. for $p=3$. In this paper we build upon the results o…
In this short note we construct a spectral, non-tiling set of size $2p$ in $(\mathbb{Z}/p\mathbb{Z})^4$, $p$ odd prime. This example complements a previous counterexample in [arXiv:1509.01090], which existed only for $p \equiv 3 \pmod{4}$. On the contrary we show that the conjecture does hold in $(\mathbb{Z}/2\mathbb{Z})^4$.
In this paper we study subsets $E$ of ${\Bbb Z}_p^d$ such that any function $f: E \to {\Bbb C}$ can be written as a linear combination of characters orthogonal with respect to $E$. We shall refer to such sets as spectral. In this context, we prove the Fuglede Conjecture in ${\Bbb Z}_p^2$ which says that $E \subset {\Bbb Z}_p^2$ is spectral if and only if $E$ tiles ${\Bbb Z}_p^2$ by translation. Arithmetic properties of the finite field Fourier transform, elementary Galois theory and combinatorial geometric properties of direction sets play the key role in the proof.
We investigate the discrete Fuglede's conjecture and Pompeiu problem on finite abelian groups and develop a strong connection between the two problems. We give a geometric condition under which a multiset of a finite abelian group has the discrete Pompeiu property. Using this description and the revealed connection we prove that Fuglede's conjecture holds for $\mathbb{Z}_{p^n q^2}$, where $p$ and $q$ are different primes. In particular, we show that every spectral subset of $\mathbb{Z}_{p^n q^2}$ tiles the group. Further, using our combinatorial methods we give a simple proof for the statement…
Fuglede's Conjecture on Cyclic Groups of Square-Free Order: The Case of Rapidly Growing Prime Factors
本文建立归纳论证，证明无穷序列无平方因子阶循环群上的Fuglede猜想，对所有这类群证明了tile-to-spectral方向，对素因子快速增长的群证明了spectral-tiling方向，填补了此前无任意多不同因子的循环群猜想成立的空白。
Fuglede's Conjecture for Cyclic Groups of Square-free Order
本文证明了无平方因子阶循环群的Fuglede猜想，给出谱集与平铺集的完整结构刻画，核心方法是素数坐标消去定理及其迭代约化。
We exhibit a subset of a finite Abelian group, which tiles the group by translation, and such that its tiling complements do not have a common spectrum (orthogonal basis for their $L^2$ space consisting of group characters). This disproves the Universal Spectrum Conjecture of Lagarias and Wang. Further, we construct a set in some finite Abelian group, which tiles the group but has no spectrum. We extend this last example to the groups $\ZZ^d$ and $\RR^d$ (for $d \ge 5$) thus disproving one direction of the Spectral Set Conjecture of Fuglede. The other direction was recently disproved by Tao.
We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T. Tao in Euclidean spaces of dimensions 5 and higher, using constructions over prime fields (in vector spaces over finite fields of prime order) and lifting them to the Euclidean setting. Over prime fields, when the dimension of the vector space is less than or equal to $2$ it has recently been proven that the Fuglede conjecture holds (see \cite{IMP15}). In t…
Steinberg conjectured in 1976 that every planar graph with no cycles of length four or five is 3-colorable. We disprove this conjecture.
The big-line-big-clique conjecture states that for all $k,\ell\geq2$ there is an integer $n$ such that every finite set of at least $n$ points in the plane contains $\ell$ collinear points or $k$ pairwise visible points. We show that this conjecture is false for infinite point sets, by constructing a countably infinite point set with no 4 collinear points and no 3 pairwise visible points.
The acyclic directed bunkbed conjecture is false
本文通过构造一个无环有向图，证明了图床猜想不成立，解决了相关猜想问题。
The weighted union-set conjecture is false
本文构造反例否证加权并集猜想，证明在满足特定权重条件下，每个元素可仅占极小权重比例。
The List Total Colouring Conjecture is False
本文构造了一个三次图，其全色数为四但列表全色数为五，从而否证了列表全染色猜想，该反例由ChatGPT 6 Astra Ultra在极少人工输入下发现。
The Aharoni--Korman conjecture is false
本文构造了一个反例，证明Aharoni和Korman在1992年提出的关于有限反链条件偏序集的猜想是错误的，并讨论了可能成立的变体。
Extending a recent breakthrough of Shitov, we prove that the chromatic number of the tensor product of two graphs can be a constant factor smaller than the minimum chromatic number of the two graphs. More precisely, we prove that there exists an absolute constant $δ>0$ such that for all $c$ sufficiently large, there exist graphs $G$ and $H$ with chromatic number at least $(1+δ)c$ for which $χ(G \times H) \le c$.
An old and well-known conjecture of Frankl and Füredi states that the Lagrangian of an $r$-uniform hypergraph with $m$ edges is maximised by an initial segment of colex. In this paper we disprove this conjecture by finding an infinite family of counterexamples for all $r \ge 4$. We also show that, for sufficiently large $t \in \mathbb{N}$, the conjecture is true in the range $\binom{t}{r} \le m \le \binom{t+1}{r} - \binom{t-1}{r-2}$.
Let $p$ be a fixed prime number, and $N$ be a large integer. The 'Inverse Conjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of a function $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constant independent of $N$, then $f$ can be non-trivially approximated by a degree $d-1$ polynomial. The conjecture is known to hold for $d=2,3$ and for any prime $p$. In this paper we show the conjecture to be false for $p=2$ and for $d = 4$, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation any polynomial of degree 3 is expon…
The Partial List Colouring Conjecture is False
本文构造了一个14顶点、列表色数为3的图，其存在2-列表分配使最多9个顶点可正确着色，从而否证了部分列表染色猜想，反例由ChatGPT 6 Astra Ultra发现并验证。
In this paper, we disprove a recent monotonicity conjecture of C. Procesi on the generating function for monotone walks on the symmetric group, an object which is equivalent to the Weingarten function of the unitary group.
In 1991, Craig Gotsman and Nathan Linial conjectured that for all $n$ and $d$, the average sensitivity of a degree-$d$ polynomial threshold function on $n$ variables is maximized by the degree-$d$ symmetric polynomial which computes the parity function on the $d$ layers of the hypercube with Hamming weight closest to $n/2$. We refute the conjecture for almost all $d$ and for almost all $n$, and we confirm the conjecture in many of the remaining cases.
O. H. Keller conjectured in 1930 that in any tiling of $\Bbb R^n$ by unit $n$-cubes there exist two of them having a complete facet in common. O. Perron proved this conjecture for $n\le 6$. We show that for all $n\ge 10$ there exists a tiling of $\Bbb R^n$ by unit $n$-cubes such that no two $n$-cubes have a complete facet in common.
We give counterexamples to the Brown-Colbourn conjecture on reliability polynomials, in both its univariate and multivariate forms. The multivariate Brown-Colbourn conjecture is false already for the complete graph K_4. The univariate Brown-Colbourn conjecture is false for certain simple planar graphs obtained from K_4 by parallel and series expansion of edges. We show, in fact, that a graph has the multivariate Brown-Colbourn property if and only if it is series-parallel.
The bunkbed conjecture is false
本文通过构造一个基于 Hollom（2024）工作的 7222 顶点平面图，首次给出 Kasteleyn 于 1985 年提出的上下铺猜想的显式反例，从而证明该猜想不成立。
One of the most important open problems in the field of graph colouring or even graph theory is the conjecture of Hadwiger. This conjecture was the inspiration for many mathematical works, one of them being the work of Füredi, Gyárfás and Simonyi in which they "risked" to conjecture the precise bound for a graph with independence number $2$ to contain a certain connected matching. We prove that their conjecture would be a corollary of Hadwiger's conjecture or equivalently if their risky conjecture would be false, then Hadwiger's conjecture would be false as well.
The matroid-based valuation conjecture of Ostrovsky and Paes Leme states that all gross substitutes valuations on $n$ items can be produced from merging and endowments of weighted ranks of matroids defined on at most $m(n)$ items. We show that if $m(n) = n$, then this statement holds for $n \leq 3$ and fails for all $n \geq 4$. In particular, the set of gross substitutes valuations on $n \geq 4$ items is strictly larger than the set of matroid based valuations defined on the ground set $[n]$. Our proof uses matroid theory and discrete convex analysis to explicitly construct a large family of c…
The Merino--Welsh conjecture is false for matroids
本文通过构造反例证明Merino–Welsh猜想在拟阵情形下不成立，具体否定了由该猜想推出的关于Tutte多项式的两个不等式。
In his book \emph{Topics in Analytic Number Theory}, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary partial fraction decomposition of the generating function for partitions of integers into at most $N$ parts exist and equal particular values that he specified. Despite being open for nearly four decades, little progress has been made toward proving or disproving the conjecture, perhaps in part due to the difficulty in actually computing the coefficients in question. In this paper, we provide a fast algorithm for calculating the Radema…
An efficient implicit representation of an $n$-vertex graph $G$ in a family $\mathcal{F}$ of graphs assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency between every pair of vertices can be determined only as a function of their codes. This function can depend on the family but not on the individual graph. Every family of graphs admitting such a representation contains at most $2^{O(n\log(n))}$ graphs on $n$ vertices, and thus has at most factorial speed of growth. The Implicit Graph Conjecture states that, conversely, every hereditary graph family with at m…
Two conjectures on the cospectrality of complete tripartite graphs are false
该研究证明了关于完全三部图K_{n,n,n}共谱性的两个猜想对所有n≥4均不成立，给出了反例图Hₙ并确定其闭式谱，还发现完全多部图约束下欧氏距离猜想渐近成立。
Kusner's conjecture is false for $p>4$
本文否证了 Kusner 关于 $\ell_p$ 度量空间中等距集最大基数的猜想对于所有 $p>4$ 成立，通过构造 $\mathbb{R}^{8m-2}$ 中 $8m$ 个点的等距集，并结合先前结果完整解决了该猜想。
The Gröbner Version of White's Conjecture is False
通过证明Fano拟阵多面体的环面理想没有二次Gröbner基，否定了拟阵论中White猜想的强版本。
Unrooted phylogenetic networks are graphs used to represent evolutionary relationships. Accurately reconstructing such networks is of great relevance for evolutionary biology. It has recently been conjectured that all phylogenetic networks with at least five leaves can be uniquely reconstructed from their subnetworks obtained by deleting a single leaf and suppressing degree-2 vertices. Here, we show that this conjecture is false, by presenting a counter example for each possible number of leaves that is at least~4. Moreover, we show that the conjecture is still false when restricted to binary …
We prove Stanley's plethysm conjecture for the $2 \times n$ case, which composed with the work of Black and List provides another proof of Foulkes conjecture for the $2 \times n$ case. We also show that the way Stanley formulated his conjecture, it is false in general, and suggest an alternative formulation.
Borsuk's conjecture states that any bounded set in R^n can be partitioned into n+1 sets of smaller diameter. It is known to be false for all n bigger or equal to 323. Here we show that Borsuk's conjecture fails in dimensions 321 and 322. (This result has been independently discovered by Hinrichs and Richter.)
The sum-product conjecture is false for real numbers
通过构造任意大的实数子集A（其元素是代数整数），证明和积猜想对于实数是错误的，并推广到多个和与积的情形。
The number of nowhere zero Z_Q flows on a graph G can be shown to be a polynomial in Q, defining the flow polynomial Φ_G(Q). According to Tutte's five-flow conjecture, Φ_G(5) > 0 for any bridgeless G.A conjecture by Welsh that Φ_G(Q) has no real roots for Q \in (4,\infty) was recently disproved by Haggard, Pearce and Royle. These authors conjectured the absence of roots for Q \in [5,\infty). We study the real roots of Φ_G(Q) for a family of non-planar cubic graphs known as generalised Petersen graphs G(m,k). We show that the modified conjecture on real flow roots is also false, by exhibiting i…
Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is replaced by any graph $H$. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erdős). This shows that Scott's conjecture is false whenever $H$ is obtained from a non-planar graph by subdividing every edge at least once. It remains …
An elementary proof of the bunkbed conjecture for forests
针对一般图上不成立的上下铺猜想，本文给出其对任意无环图（森林）成立的简短自洽初等证明，该论证具备一定教学意义。
Karasev conjectured that for any set of $3k$ lines in general position in the plane, which is partitioned into $3$ color classes of equal size $k$, the set can be partitioned into $k$ colorful 3-subsets such that all the triangles formed by the subsets have a point in common. Although the general conjecture is false, we show that Karasev's conjecture is true for lines in convex position. We also discuss possible generalizations of this result.
We show that, for any integer $\ell$ with $q-\sqrt{p} -1 \leq \ell < q-3$ where $q=p^n$ and $p>9$, there exists a multiset $M$ satisfying that $0\in M$ has the highest multiplicity $\ell$ and $\sum_{b\in M} b =0$ such that every polynomial over finite fields $\fq$ with the prescribed range $M$ has degree greater than $\ell$. This implies that Conjecture 5.1. in \cite{gac} is false over finite field $\fq$ for $p > 9$ and $k:=q-\ell -1 \geq 3$.
Progress on the union-closed conjecture and offsprings in winter 2022-2023
本文综述了2022-2023年冬季并闭猜想及其变体的研究进展，重点介绍了Justin Gilmer首次证明并闭族中最常见元素比例存在常数下界的突破性贡献。
We give constructions of some special cases of $[n,k]$ Reed-Solomon codes over finite fields of size at least $n$ and $n+1$ whose generator matrices have constrained support. Furthermore, we consider a generalisation of the GM-MDS conjecture proposed by Lovett in 2018. We show that Lovett's conjecture is false in general and we specify when the conjecture is true.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。