A Density Condition on Point Sets with Slowly-Scaling Distinct Dot Products

2025/10/16 by Gandhi, Anshula
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2510.14585

Abstract

The distinct dot products problem, a variation on the Erdős distinct distance problem, asks "Given a set P_n of n points in ℝ², what is the minimum number |D(P_n)| of distinct dot products formed between them, asymptotically?" The best proven lower-bound is |D(P_n)| \gtrsim n^2/3+7/1425, due to work by Hanson\unicodex2013Roche-Newton\unicodex2013Senger, and a recent improvement by Kokkinos. However, the slowest-scaling known constructions have |D(P_n)|∼ n, leaving quite a large gap in the bound. Finding a sublinearly-scaling construction, or disproving its existence, would narrow this gap. We provide a condition that a sequence of point configurations (P_n)_{n ∈ ℕ} must satisfy in order for |D(P_n)| to scale 'slowly' i.e. |D(P_n)| ≪ n^3/4. Namely, we prove that any such configuration must contain a point-rich line that gets arbitrarily 'dense' as the sequence progresses.

Citations

• An improved bound on the number of dot products determined by a finite point set in the plane
• Pinned Dot Product Set Estimates
• Bounds on distinct and repeated dot product trees
• Dot Product Bounds in Galois Rings
• Convexity, Superquadratic Growth, and Dot Products
• On the number of dot product chains in finite fields and rings
• Dot product chains
• Incidences and pairs of dot products
• Pairs of dot products in finite fields and rings
• Upper bounds on pairs of dot products

Related

• Dot product chains
• An improved bound on the number of dot products determined by a finite point set in the plane
• Bounds on distinct and repeated dot product trees
• Upper bounds on pairs of dot products
• Bounds on point configurations determined by distances and dot products