Research Projects

Identifying finite point configurations within thin sets (sets of zero Lebesgue measure) is an active area of research in the field of Geometric Measure Theory. A classical result due to Steinhaus says that if E, a subset of R^d, has positive Lebesgue measure, then E contains a copy of any finite point configurations. In view of Steinhaus's result, a natural question one can ask is whether ‘large’ thin sets are also rich with finite point configurations, where the largeness is measured using the Hausdorff dimension. A substantial amount of work has already been done in this direction, covering numerous classes of finite point configurations. In [BIT16], the authors investigate the richness of the class of ‘finite chains’ within sets and they prove the following. Theorem: Let d be larger than 2. For any E in Rd with dim_H(E) strictly larger than (d+1)/2, there exists a non-empty interval I such that the following holds. For any t in I, there exists a collection of distinct points x_1, x_2, ……, x_(k+1) in E such that |x_(j+1) - x_j| =t for all j. This is a generalisation of [MS99], where the authors prove the same result in the special case k=1. In the follow-up article [GIP17], the authors prove that sets of dimension larger than (d+1)/2 contain closed chains (end points of the chains are the same) with sidelengths ranging over some interval. A quick read of the proof allows one to understand that the intervals (formed by the sidelengths) mentioned in [MS99], [GIP17] and [BIT16] are all non-constructive. It is quite natural, therefore, to ask whether it is possible to explicitly find this interval, and if admittance of a particular interval is a universal property of a large collection of sets. A breakthrough result in this direction appears in [PR23], where the authors show the endpoints of the interval appearing in [MS99] depends solely on the dyadic Hausdorff content of a set, as long as the set has Hausdorff dimension so close to the ambient dimension. The exact formulation of their theorem is given in the PDF attached. The method of [PR23], which uses a spectral gap condition, promises a potential for wider usage and we believe it could lead to a universality result, quantifying the richness of chains and necklaces within thin sets. The precise formulation of the conjecture is in the attached PDF. As a next stepping stone from [PR23], we believe that our work can open doors into investigation of various other patterns within thin sets, unravelling its complexity. [BIT16] Michael Bennett, Alexander Iosevich, and Krystal Taylor. ‘Finite chains inside thin subsets of Rd’. 2016 [GIP17] Allan Greenleaf, Alex Iosevich, and Malabika Pramanik. ‘On necklaces inside thin subsets of Rd’. 2017. [PR23] Malabika Pramanik and K S Senthil Raani. ‘Distances in sparse sets of large hausdorff dimension’, 2023. [MS99] Pertti Mattila and Per Sjolin. ‘Regularity of distance measures and sets’. 1999