(If you don't need the motivation, skip it. Note, [1]-[4] are citations for links at the bottom of this post.)

Motivation: I want to find a set A⊆ℝ2 which is more non-uniform [1] and difficult to meaningfully average [2] than this set [3]. I need such a set to test my theory [4].

Suppose A⊆ℝ2 is Borel and B is a rectangle of ℝ2. In addition, suppose the Lebesgue measure on the Borel sigma 𝜎-algebra is 𝜆(.):

Question: Does there exist an explicit A such that:

• 𝜆(B\𝛽)>𝜆(𝛽)⇒𝜆(A∩(B\𝛽))