Motivic Wave Front Sets

The concept of wave front set was introduced in 1969–1970 by Sato in the hyperfunctions context [1, 34] and by Hörmander [23] in the $\mathcal C^{\infty }$ context. Howe in [25] used the theory of wave front sets in the study of Lie groups representations. Heifetz in [22] defined a notion of wave front set for distributions in the $p$-adic setting and used it to study some representations of $p$-adic Lie groups. In this article, we work in the $k\mathopen{(\!(} t \mathopen{)\!)}$-setting with $k$ a Characteristic 0 field. In that setting, balls are no longer compact but working in a definable context provides good substitutes for finiteness and compactness properties. We develop a notion of definable distributions in the framework of [13] and [14] for which we define notions of singular support and $\Lambda$-wave front sets (relative to some multiplicative subgroups $\Lambda$ of the valued field) and we investigate their behavior under natural operations like pullback, tensor product, and products of distributions.

Differences of subgroups in subgroups

We prove, in particular, that if [Formula: see text], [Formula: see text] are two arbitrary multiplicative subgroups satisfying [Formula: see text], then [Formula: see text]. Also, we obtain that for any [Formula: see text] and a sufficiently large subgroup [Formula: see text] with [Formula: see text] there is no representation [Formula: see text] as [Formula: see text], where [Formula: see text] is another subgroup, and [Formula: see text] is an arbitrary set, [Formula: see text]. Finally, we study the number of collinear triples containing in a set of [Formula: see text] and prove a variant of sum–product estimate.

Finite equal norm Parseval wavelet frames over prime fields

In the framework of wave packet analysis, finite wavelet systems are particular classes of finite wave packet systems. In this paper, using a scaling matrix on a permuted version of the discrete Fourier transform (DFT) of system generator, we derive a locally-scaled version of the DFT of system generator and obtain a finite equal-norm Parseval wavelet frame over prime fields. We also give a characterization of all multiplicative subgroups of the cyclic multiplicative group, for which the associated wavelet systems form frames. Finally, we present some concrete examples as applications of our results.