The Applied Mathematics and Physics Seminar will host Dr. Austin Anderson on Tuesday, Nov. 12, from noon-1 p.m. at IST 1015. The title of the seminar is “On the Packing Functions of Some Linear Closed Sets of Lebesgue Measure Zero.”

Abstract:

We use a characterization of Minkowski measurability to study the asymptotics of best packing on cut-out subsets of the real line with Minkowski dimension d∈(0,1). Our main result is a proof that Minkowski measurability is a sufficient condition for the existence of best packing asymptotics on monotone rearrangements of these sets. For each such set, the main result provides an explicit constant of proportionality pd, depending only on the Minkowski dimension d, that relates its packing limit and Minkowski content. We later use the Digamma function to study the limiting value of pd as d→1−. For sharpness, we use renewal theory to prove that the packing constant of the (1/2,1/3) Cantor set is less than the product of its Minkowski content and pd. We also show that the measurability hypothesis of the main theorem is necessary by demonstrating that a monotone rearrangement of the complementary intervals of the 1/3 Cantor set has Minkowski dimension d=log2/log3∈(0,1), is not Minkowski measurable, and does not have convergent first-order packing asymptotics. The aforementioned characterization of Minkowski measurability further motivates the asymptotic study of an infinite multiple subset sum problem.

• Date: Tuesday, Nov. 12
• Time: Noon-1 p.m.
• Location: IST 1015

For more information, please contact Dr. Aaron Bardall.