Wai Kiu Chan

Discriminant Bounds for Spinor Regular Ternary Quadratic Lattices

The main goals of the paper are to establish a priori bounds for the prime power divisors of the discriminants of spinor regular positive definite primitive integral ternary quadratic lattices, and to describe a procedure for determining all such lattices.

On representations of spinor genera

We determine exactly when a quadratic form is represented by a spinor genus of another quadratic form of three or four variables. We apply this to extend the embedding theorem for quaternion and also answer a question by Borovoi.

Almost universal ternary sums of triangular numbers

For any integer x, let T x denote the triangular number x(x+1)/2. In this paper we give a complete characterization of all the triples of positive integers (α, β, γ) for which the ternary sums αT x + βT y +γT z represent all but finitely many positive integers, which resolves a conjecture of Kane and Sun.

Finiteness theorems for positive definite -regular quadratic forms

An integral quadratic form f of m variables is said to be n-regular if f globally represents all quadratic forms of n variables that are represented by the genus of f. For any n ≥ 2, it is shown that up to equivalence, there are only finitely many primitive positive definite integral quadratic forms of n+3 variables that are n-regular. We also investigate similar finiteness results for almost n-regular and spinor n-regular quadratic forms. It is shown that for any n > 2, there are only finitely many equivalence classes of primitive positive definite spinor or almost n-regular quadratic forms of n + 2 variables. These generalize the finiteness result for 2-regular quaternary quadratic forms proved by Earnest (1994).

Necessary conditions for Menon difference sets

Menon difference sets have the parameters (4N2, 2N2±N,N2±N). In this paper, a necessary condition for Menon difference sets in groups of the formZ2p×Z2p×Gq, whereGq is an abelianq-group andp, q are distinct prime numbers, will be proved. We will also focus on the groupsZ2pq×Z2pq and give constraints on the magnitudes ofp andq. Finally, we show that if the groupZ6p×Z6p contains a Menon difference set, thenp=3 or 13 only.

Almost Universal Ternary Sums of Squares and Triangular Numbers

For any integer x, let T x denote the triangular number \(\frac{x(x+1)} {2}\). In this paper we give a complete characterization of all the triples of positive integers (α, β, γ) for which the ternary sums \(\alpha {x}^{2} +\beta T_{y} +\gamma T_{z}\) represent all but finitely many positive integers. This resolves a conjecture of Kane and Sun (Trans Am Math Soc 362:6425–6455, 2010, Conjecture 1.19(i)) and complete the characterization of all almost universal ternary mixed sums of squares and triangular numbers.

Nonexistence of certain perfect arrays

Abstract Perfect binary arrays are equivalent to Menon difference sets in certain groups. This note proves a theorem on the Menon difference set and interprets its consequences on the nonexistence of certain perfect arrays.

Steinhaus tiling problem and integral quadratic forms

A lattice L in R n is said to be equivalent to an integral lattice if there exists a real number r such that the dot product of any pair of vectors in rL is an integer. We show that if n ≥ 3 and L is equivalent to an integral lattice, then there is no measurable Steinhaus set for L, a set which no matter how translated and rotated contains exactly one vector in L.

Positive ternary quadratic forms with finitely many exceptions

An integral quadratic form f is said to be almost regular if f globally represents all but finitely many integers that are represented by the genus of f. In this paper, we study and characterize all almost regular positive definite ternary quadratic forms.

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watsons result to totally positive regular ternary quadratic forms over