BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Nate Hughes (University of Exeter) DTSTART:20220317T161500Z DTEND:20220317T173000Z DTSTAMP:20260423T021931Z UID:NEDNT/43 DESCRIPTION:Title: E ffective Counting and Spiralling of Lattice Approximates\nby Nate Hugh es (University of Exeter) as part of New England Dynamics and Number Theor y Seminar\n\nLecture held in Online.\n\nAbstract\nWe will prove an effecti ve version of Dirichlet’s approximation theorem\, giving the error betwe en the number of rational approximations to a real vector with denominator less than some real number T and the asymptotic growth of this count. Add itional results for linear forms can be obtained\, as well as results meas uring the direction of these approximates\, known as ‘spiralling of latt ice approximates’. These results are obtained by reformulating the numbe r-theoretic problem to the context of homogeneous spaces of unimodular lat tices. The advantage of this reformulation is that we have more tools to d eal with the problem\, such as Siegel’s mean value theorem and Rogers’ higher moment formula. The proof involves using the ergodic properties of diagonal flows on this homogeneous space to calculate the number of latti ce approximates\, bounding the second moment of the count\, then applying an effective ergodic theorem due to Gaposhkin. Particular attention is pai d to the case of primitive lattices in two-dimensions\, where Rogers’ th eorem fails. In this case\, we apply a new theorem by Kleinbock and Yu to obtain a better error term than previous results due to Schmidt.\n LOCATION:https://researchseminars.org/talk/NEDNT/43/ END:VEVENT END:VCALENDAR