Set Conjecture Research Articles

Orthogonal Fourier Analysis on domains

In this paper we go over the history of the Fuglede or Spectral Set Conjecture as it has developed over the last 30 years or so. We do not aim to be exhaustive and we do not cover important areas of development such as the results on the problem in classes of finite groups or the version of the problem that focuses on spectral measures instead of sets. The selection of the material has been strongly influenced by personal taste, history and capabilities. We are trying to be more descriptive than detailed and we point out several open questions.

Periodicity and decidability of translational tilings by rational polygonal sets

The periodic tiling conjecture asserts that if a region Σ⊂Rd tiles Rd by translations then it admits at least one fully periodic tiling. This conjecture is known to hold in R, and recently it was disproved in sufficiently high dimensions. In this paper, we study the periodic tiling conjecture for polygonal sets: bounded open sets in R2 whose boundary is a finite union of line segments. We prove the periodic tiling conjecture for any polygonal tile whose vertices are rational. As a corollary of our argument, we also obtain the decidability of tilings by rational polygonal sets. Moreover, we prove that any translational tiling by a rational polygonal tile is weakly-periodic, i.e., can be partitioned into finitely many singly-periodic pieces.

The Union-Closed Set Conjecture is True

The Union-Closed Set Conjecture is True

Log-concave polynomials III: Mason’s ultra-log-concavity conjecture for independent sets of matroids

We give a self-contained proof of the strongest version of Mason’s conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave.

The Kakeya Set conjecture over Z/NZ for general N

The Kakeya Set conjecture over Z/NZ for general N

Balancing graph Voronoi diagrams with one more vertex

AbstractLet be a graph with unit‐length edges and nonnegative costs assigned to its vertices. Given a list of pairwise different vertices , the prioritized Voronoi diagram of with respect to is the partition of in subsets so that, for every with , a vertex is in if and only if is a closest vertex to in and there is no closest vertex to in within the subset . For every with , the load of vertex equals the sum of the costs of all vertices in . The load of equals the maximum load of a vertex in . We study the problem of adding one more vertex at the end of in order to minimize the load. This problem occurs in the context of optimally locating a new service facility (e.g., a school or a hospital) while taking into account already existing facilities, and with the goal of minimizing the maximum congestion at a site. There is a brute‐force algorithm for solving this problem in time on ‐vertex ‐edge graphs. We prove a matching time lower bound–up to sub‐polynomial factors–for the special case where and , assuming the so called Hitting Set Conjecture of Abboud et al. On the positive side, we present simple linear‐time algorithms for this problem on cliques, paths and cycles, and almost linear‐time algorithms for trees, proper interval graphs and (assuming to be a constant) bounded‐treewidth graphs.

Extremal values of semi‐regular continuants and codings of interval exchange transformations

AbstractGiven a set consisting of positive integers and a k‐term partition , find the extremal denominators of the regular and semi‐regular continued fraction with partial quotients and where each occurs precisely times in the sequence . In 1983, G. Ramharter gave an explicit description of the extremal arrangements of the regular continued fraction and the minimizing arrangement for the semi‐regular continued fraction and showed that in each case the arrangement is unique up to reversal and independent of the actual values of the positive integers . However, an explicit determination of a maximizing arrangement for the semi‐regular continuant turned out to be substantially more difficult. Ramharter conjectured that as in the other three cases, the maximizing arrangement is unique (up to reversal) and depends only on the partition P and not on the actual values of the . He further verified the conjecture in the special case of a binary alphabet. In this paper, we confirm Ramharter's conjecture for sets with and give an algorithmic procedure for constructing the unique maximizing arrangement. We also show that Ramharter's conjecture fails for sets with in that the maximizing arrangement is in general neither unique nor independent of the values of the digits in . The central idea is that the extremal arrangements satisfy a strong combinatorial condition. This combinatorial condition may also be stated more or less verbatum in the context of infinite sequences on an ordered set. We show that in the context of bi‐infinite binary words, this condition coincides with the Markoff property, discovered by A. A. Markoff in 1879 in his study of minima of binary quadratic forms. We further show that this same combinatorial condition is the fundamental property which describes the orbit structure of the natural codings of points under a symmetric k‐interval exchange transformation.

A proof of the Erdős primitive set conjecture

Abstract A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the series $f(A) = \sum _{a\in A}1/(a \log a)$ is uniformly bounded over all choices of primitive sets A. In 1986, he asked if this bound is attained for the set of prime numbers. In this article, we answer in the affirmative. As further applications of the method, we make progress towards a question of Erdős, Sárközy and Szemerédi from 1968. We also refine the classical Davenport–Erdős theorem on infinite divisibility chains, and extend a result of Erdős, Sárközy and Szemerédi from 1966.

On holomorphic foliations admitting invariant CR manifolds

We study holomorphic foliations of codimension $k\geq 1$ on a complex manifold $X$ of dimension $n+k$ from the point of view of the exceptional minimal set conjecture. For $n\geq 2$ we show in particular that if the holomorphic normal bundle $N_{\mathcal{F}}$ is Griffiths positive, then the foliation does not admit a compact invariant set that is a complete intersection of $k$ smooth real hypersurfaces in $X$.

Translated sums of primitive sets

The Erd\H{o}s primitive set conjecture states that the sum $f(A) = \sum_{a\in A}\frac{1}{a\log a}$, ranging over any primitive set $A$ of positive integers, is maximized by the set of prime numbers. Recently Laib, Derbal, and Mechik proved that the translated Erd\H{o}s conjecture for the sum $f(A,h) = \sum_{a\in A}\frac{1}{a(\log a+h)}$ is false starting at $h=81$, by comparison with semiprimes. In this note we prove that such falsehood occurs already at $h= 1.04\cdots$, and show this translate is best possible for semiprimes. We also obtain results for translated sums of $k$-almost primes with larger $k$.

On the structure of spectral and tiling subsets of cyclic groups

AbstractThe purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture [9] in one dimension, which states that a bounded measurable subset of$\mathbb {R}$accepts an orthogonal basis of exponentials if and only if it tiles$\mathbb {R}$by translations. This conjecture is strongly connected to its discrete counterpart, namely that, in every finite cyclic group, a subset is spectral if and only if it is a tile. The tools presented herein are refinements of recent ones used in the setting of cyclic groups; the structure of vanishing sums of roots of unity [20] is a prevalent notion throughout the text, as well as the structure of tiling subsets of integers [1]. We manage to prove the conjecture for cyclic groups of order$p^{m}q^{n}$, when one of the exponents is$\leq 6$or when$p^{m-2}<q^{4}$, and also prove that a tiling subset of a cyclic group of order$p_{1}^{m}p_{2}\dotsm p_{n}$is spectral.

Proof of the Kakeya set conjecture over rings of integers modulo square-free N

A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ is a set containing a line in each direction. We show that, when $N$ is any square-free integer, the size of the smallest Kakeya set in $(\mathbb{Z}/N\mathbb{Z})^n$ is at least $C_{n,\epsilon} N^{n - \epsilon}$ for any $\epsilon$ -- resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime $N$. We also show that the case of general $N$ can be reduced to lower bounding the $\mathbb{F}_p$ rank of the incidence matrix of points and hyperplanes over $(\mathbb{Z}/p^k\mathbb{Z})^n$.

On the Critical Exponent for k-Primitive Sets

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erd\H{o}s proved in 1935 that the weighted sum $\sum1/(n \log n)$ for $n$ ranging over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that $\sum n^{-\lambda}$ over a primitive set is maximized by the primes if and only if $\lambda$ is at least the critical exponent $\tau_1 \approx 1.14$. A set is $k$-primitive if no member divides any product of up to $k$ other distinct members. One may similarly consider the critical exponent $\tau_k$ for which the primes are maximal among $k$-primitive sets. In recent work the authors showed that $\tau_2 < 0.8$, which directly implies the Erd\H{o}s conjecture for 2-primitive sets. In this article we study the limiting behavior of the critical exponent, proving that $\tau_k$ tends to zero as $k\to\infty$.

An asymptotic version of the prime power conjecture for perfect difference sets

We show that the number of positive integers $$n\le N$$ such that $${\mathbb {Z}}/(n^2+n+1){\mathbb {Z}}$$ contains a perfect difference set is asymptotically $$\frac{N}{\log {N}}$$ .

The strength of Menger's conjecture

Menger conjectured that subsets of R with the Menger property must be σ-compact. While this is false when there is no restriction on the subsets of R, for projective subsets it is known to follow from the Axiom of Projective Determinacy, which has considerable large cardinal consistency strength. We note that in fact, Menger's conjecture for projective sets has consistency strength of only an inaccessible cardinal.

Probabilistic existence of large sets of designs

A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been introduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a t-(n,k,λ) combinatorial design with tiny, yet positive, probability.The proof method of KLP is adapted to show the existence of large sets of designs and similar combinatorial structures as follows. We modify the random choice and the analysis to show that, under the same conditions, not only does a t-(n,k,λ) design exist but, in fact, with positive probability there exists a large set of such designs — that is, a partition of the set of k-subsets of [n] into t-(n,k,λ) designs. Specifically, using the probabilistic approach derived herein, we prove that for all sufficiently large n, large sets of t-(n,k,λ) designs exist whenever k>12t and the necessary divisibility conditions are satisfied. This resolves the existence conjecture for large sets of designs for all k>12t.

Equi‐distributed property and spectral set conjecture on Zp2×Zp

In this paper, we show an equi-disctributed property in $2$-dimensional finite abelian groups $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$ where $p$ is a prime number. By using this equi-disctributed property, we prove that Fuglede's spectral set conjecture holds on groups $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$, namely, a set in $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$ is a spectral set if and only if it is a tile.

Self‐conformal sets with positive Hausdorff measure

We investigate the Hausdorff measure and content on a class of quasi self-similar sets that include, for example, graph-directed and sub self-similar and self-conformal sets. We show that any Hausdorff measurable subset of such a set has comparable Hausdorff measure and Hausdorff content. In particular, this proves that graph-directed and sub self-conformal sets with positive Hausdorff measure are Ahlfors regular, irrespective of separation conditions. When restricting to self-conformal subsets of the real line with Hausdorff dimension strictly less than one, we additionally show that the weak separation condition is equivalent to Ahlfors regularity and its failure implies full Assouad dimension. In fact, we resolve a self-conformal extension of the dimension drop conjecture for self-conformal sets with positive Hausdorff measure by showing that its Hausdorff dimension falls below the expected value if and only if there are exact overlaps.

Viterbo’s Conjecture for Certain Hamiltonians in Classical Mechanics

We study some particular cases of Viterbo's conjecture relating volumes of convex bodies and actions of closed characteristics on their boundaries, focusing on the case of a Hamiltonian of classical mechanical type, splitting into summands depending on the coordinates and the momentum separately. We manage to establish the conjecture for sublevel sets of convex $2$-homogeneous Hamiltonians of this kind in several particular cases. We also discuss open cases of this conjecture.

Extremal Union-Closed Set Families

A family of finite sets is called union-closed if it contains the union of any two sets in it. The Union-Closed Sets Conjecture of Frankl from 1979 states that each union-closed family contains an element that belongs to at least half of the members of the family. In this paper, we study structural properties of union-closed families. It is known that under the inclusion relation, every union-closed family forms a lattice. We call two union-closed families isomorphic if their corresponding lattices are isomorphic. Let $${{\mathcal {F}}}$$ be a union-closed family and $$\bigcup _{F\in {\mathcal {F}}} F$$ be the universe of $${\mathcal {F}}$$. Among all union-closed families isomorphic to $${{\mathcal {F}}}$$, we develop an algorithm to find one with a maximum universe, and an algorithm to find one with a minimum universe. We also study properties of these two extremal union-closed families in connection with the Union-Closed Set Conjecture. More specifically, a lower bound of sizes of sets of a minimum counterexample to the dual version of the Union-Closed Sets Conjecture is obtained.