Scaling scenery and the distance set problem

Given a set E ⊂ Rd, the distance set of E is the set of all distances realised by pairs of points in E. D(E) = {|x − y | : x, y ∈ E}. It is interesting to compare the ‘sizes ’ of E and D(E): Question: Suppose E has cardinality n, what is the minimum possible cardinality of D(E)? Jonathan Fraser The distance set problem Distance sets Given a set E ⊂ Rd, the distance set of E is the set of all distances realised by pairs of points in E. D(E) = {|x − y | : x, y ∈ E}. It is interesting to compare the ‘sizes ’ of E and D(E): Question: Suppose E has cardinality n, what is the minimum possible cardinality of D(E)? Jonathan Fraser The distance set problem Distance sets Given a set E ⊂ Rd, the distance set of E is the set of all distances realised by pairs of points in E. D(E) = {|x − y | : x, y ∈ E}. It is interesting to compare the ‘sizes ’ of E and D(E): Question: Suppose E has cardinality n, what is the minimum possible cardinality of D(E)? Jonathan Fraser The distance set problem Distance sets Given a set E ⊂ Rd, the distance set of E is the set of all distances realised by pairs of points in E. D(E) = {|x − y | : x, y ∈ E}. It is interesting to compare the ‘sizes ’ of E and D(E): Question: Suppose E has cardinality n, what is the minimum possible cardinality of D(E)? Jonathan Fraser The distance set problem Distance sets Let g(n) = min{|D(E) | : |E | = n}