arXiv subjects
Publications and source records attributed to Andrew Tung.
For symplectic and even orthogonal groups over a $p$-adic field, we determine the number of local Arthur packets containing the local theta lift of a tempered representation at the first occurrence in the going-up tower. These counts show that the local theta lifts may lie in many more local Arthur packets than those predicted by the Adams conjecture.
We determine the number of local Arthur packets containing a certain fixed tempered representation for classical $p$-adic groups. More specifically, given a tempered extended multi-segment supported in the integers, we determine a count for all extended multi-segments which arise from it through applications of the operators arising from the theory of intersections of local Arthur packets.
We examine a unique construction of the real numbers which proceeds directly from the integers using approximately linear-endomorphisms with finite error, called near-endomorphisms. In this paper, we show that the set of near-endomorphisms forms a complete ordered field isomorphic to the reals. Moreover, we show that there are uncountably many near-endomorphisms without reference to the reals. We then investigate a natural extension of near-endomorphisms, which we call quasi-homomorphisms, to other abelian groups. Extending prior results about the construction of the $p$-adic numbers and the rational adele ring, we find the ring of near-endomorphisms of certain localizations of the integers, and suggest further directions for exploration.
The secret-key rate measures the rate at which Alice and Bob can extract secret bits from sampling a joint probability distribution, unknown to an eavesdropper Eve. The secret-key rate has been bounded above by the intrinsic information and reduced intrinsic information. However, we prove that the reduced intrinsic information is 0 if and only if the intrinsic information is 0. This result implies that at least one of the following two conjectures is false: bound secrecy exists, or the reduced intrinsic information equals the secret-key rate. We give an explicit construction of an information-erasing binarization for a candidate for bound secrecy. We then introduce some approaches for proving the existence of bound secrecy, such as reducing the channel space, linearly transforming Bob's map, and perturbing a channel for Eve.