We classify the sets of four lattice points that all lie on a short arc of a circle that has its center at the origin; specifically on arcs of length \( tR^{1/3}\) on a circle of radius \( R\), for any given \( t>0\).

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We find the size of the smallest subset of the set of integer lattice points, such that every element of a given rectangular grid is visible from our subset, which answers a question of Paul Erdos.

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This is a written version of my lecture at the Journée Arithmétique in Edinburgh, in which I describe how, in joint work with K. Soundararajan and Antal Balog, we have developed the notion of "pretentiousness" to help us better understand several key questions in analytic number theory.

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