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Register a new userDistinct spreads in vector spaces over finite fields
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In this short note, we study the distribution of spreads in a point set $mathcal{P} subseteq mathbb{F}_q^d$, which are analogous to angles in Euclidean space. More precisely, we prove that, for any $varepsilon > 0$, if $|mathcal{P}| geq (1+varepsilon) q^{lceil d/2 rceil}$, then $mathcal{P}$ generates a positive proportion of all spreads. We show that these results are tight, in the sense that there exist sets $mathcal{P} subset mathbb{F}_q^d$ of size $|mathcal{P}| = q^{lceil d/2 rceil}$ that determine at most one spread.
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We prove that a sufficiently large subset of the $d$-dimensional vector space over a finite field with $q$ elements, $ {Bbb F}_q^d$, contains a copy of every $k$-simplex. Fourier analytic methods, Kloosterman sums, and bootstrapping play an important role.
We say that $M$ and $S$ form a textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $min M$ and $sin S$, while $0$ has no such representation. The splitting is called {it nonsingular} if $gcd(|G|, a) = 1$ for any $ain M$. In this paper, we focus our study on nonsingular splittings of cyclic groups. We introduce a new notation --direct KM logarithm and we prove that if there is a prime $q$ such that $M$ splits $mathbb{Z}_q$, then there are infinitely many primes $p$ such that $M$ splits $mathbb{Z}_p$.
We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${mathbb F}_q$, the finite field with q elements, by $A cdot A+... +A cdot A$, where A is a subset ${mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting due to the influence of the arithmetic. On the positive side, we obtain good exponents for the Erdos -Falconer distance problem for subsets of the unit sphere in $mathbb F_q^d$ and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher dimensional vector spaces over general finite fields.
An orthomorphism over a finite field $mathbb{F}_q$ is a permutation $theta:mathbb{F}_qmapstomathbb{F}_q$ such that the map $xmapstotheta(x)-x$ is also a permutation of $mathbb{F}_q$. The degree of an orthomorphism of $mathbb{F}_q$, that is, the degree of the associated reduced permutation polynomial, is known to be at most $q-3$. We show that this upper bound is achieved for all prime powers $q otin{2, 3, 5, 8}$. We do this by finding two orthomorphisms in each field that differ on only three elements of their domain. Such orthomorphisms can be used to construct $3$-homogeneous Latin bitrades.
We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but~18 known SRG parameters on $v leq 1300$ vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzons bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzons bound cannot be attained in a finite orthogonal geometry of dimension~5.
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