On Directions Determined by Subsets of Vector Spaces over Finite Fields
-
Alex Iosevich
,
Hannah Morgan
Abstract
We prove that if a subset of a d-dimensional vector space over a finite field with q elements has more than qd–1 elements, then it determines all the possible directions. We obtain a complete characterization if the size of the set is ≥ qd – 1. If a set has more than qk elements, it determines a k-dimensional set of directions. We prove stronger results for sets that are sufficiently random. This result is best possible as the example of a k-dimensional hyperplane shows. We can view this question as an Erdős type problem where a sufficiently large subset of a vector space determines a large number of configurations of a given type. For discrete subsets of 
, this question has been previously studied by Pach, Pinchasi and Sharir.
© de Gruyter 2011
Articles in the same Issue
- Completely Multiplicative Automatic Functions
- The k-Periodic Fibonacci Sequence and an Extended Binet's Formula
- Robin's Theorem, Primes, and a New Elementary Reformulation of the Riemann Hypothesis
- Reducing the Erdős–Moser Equation 1n + 2n + ⋯ + kn = (k + 1)n Modulo k and k2
- On Some Conjectures Concerning Stern's Sequence and Its Twist
- Number of Weighted Subsequence Sums with Weights in {1, –1}
- Binomial Coefficient – Harmonic Sum Identities Associated to Supercongruences
- Euler's Pentagonal Number Theorem Implies the Jacobi Triple Product Identity
- On Directions Determined by Subsets of Vector Spaces over Finite Fields
- A Remark on a Paper of Luca and Walsh
- On the Tennis Ball Problem
- On the Conditioned Binomial Coefficients
- Convolution and Reciprocity Formulas for Bernoulli Polynomials
- Counting Finite Languages by Total Word Length
Articles in the same Issue
- Completely Multiplicative Automatic Functions
- The k-Periodic Fibonacci Sequence and an Extended Binet's Formula
- Robin's Theorem, Primes, and a New Elementary Reformulation of the Riemann Hypothesis
- Reducing the Erdős–Moser Equation 1n + 2n + ⋯ + kn = (k + 1)n Modulo k and k2
- On Some Conjectures Concerning Stern's Sequence and Its Twist
- Number of Weighted Subsequence Sums with Weights in {1, –1}
- Binomial Coefficient – Harmonic Sum Identities Associated to Supercongruences
- Euler's Pentagonal Number Theorem Implies the Jacobi Triple Product Identity
- On Directions Determined by Subsets of Vector Spaces over Finite Fields
- A Remark on a Paper of Luca and Walsh
- On the Tennis Ball Problem
- On the Conditioned Binomial Coefficients
- Convolution and Reciprocity Formulas for Bernoulli Polynomials
- Counting Finite Languages by Total Word Length
