Abstract
We prove a discretized sum-product theorem for representations of Lie groups whose Jordan–Hölder decomposition does not contain the trivial representation. This expansion result is used to derive a product theorem in perfect Lie groups.
Highlights
Throughout this paper, G will denote a connected real Lie group, endowed with a left-invariant Riemmanian metric
For A ⊂ G, X ⊂ V and s ≥ 1, we denote by A, X s the set of elements in V that can be obtained as combinations of sums, differences and products of at most s elements from A and X
1.2 Product theorem in perfect Lie groups
Summary
Throughout this paper, G will denote a connected real Lie group, endowed with a left-invariant Riemmanian metric. For x ∈ G and ρ > 0, we denote by BG(x, ρ) the ball of center x and radius ρ in G. For A ⊂ G and ρ > 0, A(ρ) stands for the ρ-neighborhood of A and N (A, ρ) stands for the covering number of A by ρ-balls, i.e. N (A, ρ) = min N ∈ N | ∃ x1, . The same notation is used for other metric spaces
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