Tight wavelet frame sets in finite vector spaces

Let $q\ge 2$ be an integer, and ${\mathbb{F}}_q^d$, $d\ge 1$, be the vector space over the cyclic space ${\mathbb{F}}_q$. The purpose of this paper is two-fold. First, we obtain sufficient conditions on $E?{\mathbb{F}}_q^d$ such that the inverse Fourier transform of $1_E$ generates a tight wavelet frame in $L^2({\mathbb{F}}_q^d)$. We call these sets (tight) wavelet frame sets. The conditions are given in terms of multiplicative and translational tilings, which is analogous with Theorem 1.1 ([20]) by Wang in the setting of finite fields. In the second part of the paper, we exhibit a constructive method for obtaining tight wavelet frame sets in ${\mathbb{F}}_q^d$, $d\ge 2$, q an odd prime and $q\equiv 3$ (mod 4).     

Linear sets in finite projective spaces

In this paper linear sets of finite projective spaces are studied and the “dual” of a linear set is introduced. Also, some applications of the theory of linear sets are investigated: blocking sets in Desarguesian planes, maximum scattered linear sets, translation ovoids of the Cayley Hexagon, translation ovoids of orthogonal polar spaces and finite semifields. Besides “old” results, new ones are proven and some open questions are discussed.     

Kakeya sets in finite affine spaces

We construct Kakeya sets in AG(n,q), where q is even and n?2, whose points are zeros of a polynomial of degree q.     

Nathanson heights in finite vector spaces

Let p be a prime, and let Zp denote the field of integers modulo p. The Nathanson height of a point is the sum of the least nonnegative integer representatives of its coordinates. The Nathanson height of a subspace is the least Nathanson height of any of its nonzero points. In this paper, we resolve a quantitative conjecture of Nathanson [M.B. Nathanson, Heights on the finite projective line, Int. J. Number Theory, in press], showing that on subspaces of of codimension one, the Nathanson height function can only take values about . We show this by proving a similar result for the coheight on subsets of Zp, where the coheight of A⊆Zp is the minimum number of times A must be added to itself so that the sum contains 0. We conjecture that the Nathanson height function has a similar constraint on its range regardless of the codimension, and produce some evidence that supports this conjecture.     

Vector packing in finite dimensional vector spaces

Let m be an integer ? 2. The effect of crowding m unit vectors x₁,…,x_m into the real Euclidean space R_n of n dimensions is investigated. In particular, several upper bounds for the quantity min_i≠j∥x_i ? x_j∥ are obtained. These are simpler than any previously known and, at least in some cases, almost as sharp. The results have application to the so-called maximum-dispersal (or “misanthrope”) problem, an open problem recently popularized by Klee.     

Monochromatic affine lines in finite vector spaces

Let d(k, q) be the smallest positive integer d such that if the d-dimensional vector space over the q-element field is k-colored, there exists a monochromatic affine line. It is shown that d(2, 4) = 3 and d(3, 3) = 4.     

Families of intersecting finite vector spaces

The following theorem is proved. It is a generalization of the problem for finite vector spaces analogous to a theorem of Kleitman for finite sets.Let V be an n-dimensional vector space over a finite field F of q elements. Suppose we have two collections, one consisting of k- and the other of m-dimensional subspaces of V with the property that the intersection of each member of one with each member of the other has dimension no less than r.Then if n ? k + m + 2 or n ? k + m + 1 and q ? 3, there are either no more than $\text{[}\text{n?r}\text{k?r}\text{]}{}_{\mathrm{q}}$ members in the first family or fewer than $\text{[}\text{n?r}\text{m?r}\text{]}{}_{\mathrm{q}}$ members in the second.The method used leads to a similar result for sets, provided that n ? r + (r + 1)(k ? r)(m ? r + 1) with k ? m.     

On intriguing sets of finite symplectic spaces

Some constructions of intriguing sets of finite symplectic spaces are provided. In particular an affirmative answer to an existence question about small tight sets posed in De Beule et al. (Des Codes Cryptogr 50(2):187–201, 2009) is given.     

Blocking sets and partial spreads in finite projective spaces

A t-blocking set in the finite projective space PG(d, q) with dt+1 is a set of points such that any (d–t)-dimensional subspace is incident with a point of and no t-dimensional subspace is contained in . It is shown that | |q ^t +...+1+q ^t–1q and the examples of minimal cardinality are characterized. Using this result it is possible to prove upper and lower bounds for the cardinality of partial t-spreads in PG(d, q). Finally, examples of blocking sets and maximal partial spreads are given.     

Partitions of finite vector spaces into subspaces

Let V_n(q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V_n(q) is a partition of V_n(q) if every nonzero element of V_n(q) is contained in exactly one element of . Suppose there exists a partition of V_n(q) into x_i subspaces of dimension n_i, 1 ≤ i ≤ k. Then x₁, …, x_k satisfy the Diophantine equation . However, not every solution of the Diophantine equation corresponds to a partition of V_n(q). In this article, we show that there exists a partition of V_n(2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7x + 3y = 2ⁿ ? 1 and y ≠ 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of V_n(q) induce uniformly resolvable designs on qⁿ points. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 329–341, 2008     

Threshold functions for incidence properties in finite vector spaces

The main purpose of this paper is to provide threshold functions for the events that a random subset of the points of a finite vector space has certain properties related to point-flat incidences. Specifically, we consider the events that there is an ℓ-rich m-flat with regard to a random set of points in ${\mathbb{F}}_q^n$, the event that a random set of points is an m-blocking set, and the event that there is an incidence between a random set of points and a random set of m-flats. One of our key ingredients is a stronger version of a recent result obtained by Chen and Greenhill (2021).     

The relative Chebyshev centers of finite sets in geodesic spaces

In the present paper we estimate variation in the relative Chebyshev radius R _W (M), where M and W are nonempty bounded sets of a metric space, as the sets M and W change. We find the closure and the interior of the set of all N-nets each of which contains its unique relative Chebyshev center, in the set of all N-nets of a special geodesic space endowed by the Hausdorff metric. We consider various properties of relative Chebyshev centers of a finite set which lie in this set.     

Paley-like graphs over finite fields from vector spaces

Motivated by the well-known Paley graphs over finite fields and their generalizations, in this paper we explore a natural multiplicative-additive analogue of such graphs arising from vector spaces over finite fields. Namely, if $n\ge 2$ and $U⊊{\mathbb{F}}_{q^n}$ is an ${\mathbb{F}}_q$-vector space, $G_U$ is the (undirected) graph with vertex set $V(G_U)={\mathbb{F}}_{q^n}$ and edge set $E(G_U)=\{(a,b)\in {\mathbb{F}}_{q^n}^2|a\ne b,ab\in U\}$. We describe the structure of an arbitrary maximal clique in $G_U$ and provide bounds on the clique number $\omega (G_U)$ of $G_U$. In particular, we compute the largest possible value of $\omega (G_U)$ for arbitrary q and n. Moreover, we obtain the exact value of $\omega (G_U)$ when $U⊊{\mathbb{F}}_{q^n}$ is any ${\mathbb{F}}_q$-vector space of dimension $d_U\in \{1,2,n-1\}$.