Two objects with zero volume can be subtracted from each other to create a solid, three-dimensional space.
March 31, 2026
Original Paper
Thick Difference Sets of Haar Null Compact Sets in Locally Compact Groups
arXiv · 2603.27479
The Takeaway
In everyday logic, combining two things that take up no space should still result in no space. This proof shows that 'thin' sets of points can be arranged so that the difference between them fills an entire region, effectively creating substance out of a void.
From the abstract
Let \(G\) be a non-discrete, locally compact group with Haar measure \(m\). We prove that there exists a compact set \(K \subset G\) with \(m(K)=0\) such that \(KK^{-1}\) contains a neighborhood of the identity. Moreover, such a set may be constructed inside any prescribed neighborhood of the identity.