Spectral gap in the group of affine transformations over prime fields

Lindenstrauss, Elon; Varjú, Péter P.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 5, pp. 969-993.

Nous étudions les marches aléatoires sur les groupes ${𝔽}_p^d⋊{\mathrm{SL}}_{\mathrm{d}}\left({𝔽}_{\mathrm{p}}\right)$. Nous estimons le trou spectral en fonction du trou spectral de la projection sur la partie linéaire ${\mathrm{SL}}_{\mathrm{d}}\left({𝔽}_{\mathrm{p}}\right)$. Ce problème est motivé par son analogue dans le groupe ${ℝ}^d⋊\mathrm{SO}\left(\mathrm{d}\right)$, qui a des applications à la régularité des mesures auto-similaires.

We study random walks on the groups ${𝔽}_p^d⋊{\mathrm{SL}}_{\mathrm{d}}\left({𝔽}_{\mathrm{p}}\right)$. We estimate the spectral gap in terms of the spectral gap of the projection to the linear part ${\mathrm{SL}}_{\mathrm{d}}\left({𝔽}_{\mathrm{p}}\right)$. This problem is motivated by an analogue in the group ${ℝ}^d⋊\mathrm{SO}\left(\mathrm{d}\right)$, which have application to smoothness of self-similar measures.

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