Maximising the number of solutions to a linear equation in a set of integers

Abstract

Given a linear equation of the form a 1 x 1 + a 2 x 2 + a 3 x 3 = 0 with integer coefficients a i , we are interested in maximising the number of solutions to this equation in a set S ⊆ Z , for sets S of a given size. We prove that, for any choice of constants a 1 , a 2 and a 3 , the maximum number of solutions is at least ( 1 12 + o ( 1 ) ) | S | 2 . Furthermore, we show that this is optimal, in the following sense. For any ε > 0 , there are choices of a 1 , a 2 and a 3 , for which any large set S of integers has at most ( 1 12 + ε ) | S | 2 solutions. For equations in k ⩾ 3 variables, we also show an analogous result. Set σ k = ∫ − ∞ ∞ ( sin π x π x ) k d x . Then, for any choice of constants a 1 , … , a k , there are sets S with at least ( σ k k k − 1 + o ( 1 ) ) | S | k − 1 solutions to a 1 x 1 + ⋯ + a k x k = 0 . Moreover, there are choices of coefficients a 1 , … , a k for which any large set S must have no more than ( σ k k k − 1 + ε ) | S | k − 1 solutions, for any ε > 0 .

Locations

• Bulletin of the London Mathematical Society - View - PDF
• arXiv (Cornell University) - View - PDF
• DataCite API - View