Average Bound Toward the Generalized Ramanujan Conjecture and Its Applications on Sato–Tate Laws for GL(n)

We give the 1st non-trivial estimate for the number of $GL(n)$ ($n\ge 3$) Hecke–Maass forms whose Satake parameters at any given prime $p$ fail the Generalized Ramanujan Conjecture and study some applications on the (vertical) Sato–Tate laws.

Lifted Hybrid Variational Inference

Lifted inference algorithms exploit model symmetry to reduce computational cost in probabilistic inference. However, most existing lifted inference algorithms operate only over discrete domains or continuous domains with restricted potential functions. We investigate two approximate lifted variational approaches that apply to domains with general hybrid potentials, and are expressive enough to capture multi-modality. We demonstrate that the proposed variational methods are highly scalable and can exploit approximate model symmetries even in the presence of a large amount of continuous evidence, outperforming existing message-passing-based approaches in a variety of settings. Additionally, we present a sufficient condition for the Bethe variational approximation to yield a non-trivial estimate over the marginal polytope.

Sums of Dilates in Groups of Prime Order

We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if t is an integer different from 0, 1 or −1 and if ⊂ ℤ/pℤ is not too large (with respect to p), then | + t ⋅ | is significantly larger than 2|| (unless |t| = 3). In the important case |t| = 2, we obtain for instance | + t ⋅ | ≥ 2.08 ||−2.

Trigonometric sums of Heilbronn's type

In problems of additive number theory one frequently needs to obtain a non-trivial estimate for the absolute value of a trigonometric sum of the typewhere f(X) ε ℤ [X] and 1 ≤ m ε ℤ. The general procedure is first to reduce the estimation to the case where m is a prime power, by means of the Chinese Remainder Theorem. The case m = pr (p prime) can often be reduced to that of a lower power of p, by a substitution of the type x = u + vps (where 0 ≤ u < ps and 0 ≤ v < pr-s), followed by the use of a p-adic Taylor expansion f(u+psv) = f(u) +psvf′(u) +…. Frequently this gives T(f, pr) = 0 when r ≥ 2, or at least allows one to reduce to the case m = p. In the latter case an appeal to Weil's estimateusually gives a good estimate for (0.1), at least if deg f = o(√p).