The Erdős-Anning Theorem states that an integer distance set in the Euclidean plane must have all of its points on a single line or is finite. However, this is not true if we consider area sets. That is, if \((x_1,y_1)\) and \((x_2,y_2)\) are any two vectors contained in the integer lattice, then the area of the parallelogram determined by the two vectors is an integer, showing that the points do not have to lie on a line. We prove a finite field version of these results for \(d=2\) and \(d=3\), showing that if \(E \subset \Bbb{F}_q^d, q=p^2\), where \(p\) is an odd prime and the distance set of \(E\) is \(\Bbb{F}_p\), then the size of \(E\) is at most \(p^d\). Furthermore, we prove that if the area set of \(E\) is a subset of \(\Bbb{F}_p\), then the size of \(E\) is at most \(p^2\) in two dimensions.
Let Fn denote the n-th Fibonacci number defined by Fn = Fn − 1 + Fn − 2 if n ≥ 2, with F0 = 0 and F1 = 1. In this paper, we find determinant identities for several Toeplitz–Hessenberg matrices whose nonzero entries are derived from the sequence kn + m for various fixed m, where kn = Fn − 1. These results may be obtained algebraically as special cases of more general formulas involving the Horadam numbers and the generating functions for the associated sequences of determinants. Equivalent multi-sum identities featuring sums of products of kn terms with multinomial coefficients may be given, which follow from Trudi’s formula. Connections are made to several OEIS entries that have arisen previously in other contexts, perhaps most notably the Padovan number sequence. Finally, we provide combinatorial proofs of our identities involving kn by enumerating (or finding the sum of signs of) various classes of tilings containing squares, dominos, trominos and a special type of tile which can be of arbitrary length.
