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Abstract

A classical question in discrete geometry asks: how many times can a given angle α ∈ (0, π) be observed among the triples of a set of n points in the plane? In 1992, Pach and Sharir proved that the maximal number of dis- crete repeated angles is Θ(n^2 log n) for most choices of α, including π/2. In this thesis, we study an approximate version of the repeated angles problem, wherein we relax our count to include triples which are ‘δ-close’ to deter- mining angle α. After the minimum necessary assumptions to make sense of this so-called fuzzy variant, we provide a construction yielding Ω(n^(5/2)) repeated angles and prove a general upper bound of O(δ^(−1/3) n^(7/3) log n) re- peated angles. We then show that the upper bound can be applied to study repeated angles in the continuum, bounding the Hausdorff dimension of the set of repeated angles in a ground set P by a function of the Minkowski dimension of P. Finally, we propose methods to close the gap between our lower and upper bounds and discuss improvements to the bounds under stronger assumptions on the distribution of the points. This thesis is meant to serve as a guiding example of the broader utility of discretized or fuzzy geometry to connect the study of geometric pattern problems in the discrete and continuum settings.