*[Fair enough; I’ve reworded the lemma. -T.]*

*[Corrected, thanks – there was a missing step in the proof in which one normalised f and g to have sup norm 1. -T.]*

the Holder inequality CANNOT be used since it is in the reverse order.

It seems you prove something slightly weaker than the inequality wanted.

in fact . ]]>

Shortly said, what I believe is that the result of Breuillard and Green

arXiv:0906.3598, saying that approximate groups in Carnot groups are controlled by nilboxes, is true (in some asymptotic regimes maybe) in a more general situation, i.e. without supposing that the ambient group is nilpotent or even without any ambient group hypothesis (other than the ones which can be turned into hypotheses on the approximate group itself).

My hope is that it is possible to prove this from the start, without invoking a Gromov theorem & Hilbert 5 problem (in order to construct the ambient somehow). One way would be to prove that there are some families of dilations (i.e. some “homomorphisms” of “nilboxes”) and only in the end prove by abstract nonsense that in some limit the “nilboxes” have to be really nilboxes and so on.

Or, there is another thing maybe, turn the reasoning upside up and use Breuillard-Green like result to prove that any approximate group which is roughly equivalent with a nilbox has some dilations (in a rough sense), then quantify the complexity of such objects by estimating how much the use of said dilations simplify the word problem.

I have a strong feeling that your program on H5 problem and approximate groups is close to these ideas.

]]>Trivial example:

Take with the Heisenberg group operation and . The set is an approximate group and, by your definition, for $n \in \mathbb{N}^{*}$, that $\frac{1}{n} A(N) = \left\{ (a,b,c) \in G \mbox{ : } \mid a \mid \leq \frac{N}{n}, \mid b \mid \leq \frac{N}{n} , \mid c \mid \leq \frac{N^{2}}{n} \right\}$, which is far from . That is why, in the limit “this gives Euclidean …”.

Define $n \cdot (a,b,c) =(n a, n b, n^{2} c)$, then $n \cdot A(\frac{N}{n}) = A(N)$. This is the “true” self-similarity in this case. As you see, the definition of it is not easy to write, in the sense that if I give you an element of , written as a word with letters from , it is far from obvious to tell how to produce the element of such that is closest to .

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