In this talk we will discuss the paper by Luca Brandolini and Giancarlo Travaglini, with a particular focus on Sections 2 and 3. Our goal is to develop upper bounds for the lattice point discrepancy in dimension two, that is, the deviation between the number of lattice points contained in the dilated body and its area, focusing on planar convex bodies whose boundary coincides locally with the graph of (the absolute value of) a power function. We will see that the Fourier series representation of the discrepancy function involves the Fourier transform of the indicator function  of the dilated body. The first step is to obtain an upper bound for this Fourier transform using chords perpendicular to lines in a given direction. This geometric description provides a key tool for establishing precise upper bounds for the Fourier transform of the indicator function (Section 2 of the paper) and, ultimately, for its L^p (spherical) average decay. The aim of the talk is to present the underlying ideas and to discuss the proofs in full detail.

 L. Brandolini,  G. Travaglini. "Fourier analytic techniques for lattice point discrepancy." Discrepancy theory 26 (2020).