Orthogonal sequences and regularity of embeddings into finite-dimensional spaces

Abstract

This paper focuses on the regularity of linear embeddings of finite-dimensional subsets of Hilbert and Banach spaces into Euclidean spaces. We study orthogonal sequences in a Hilbert space H, whose elements tend to zero, and similar sequences in the space c 0 of null sequences. The examples studied show that the results due to Hunt and Kaloshin (Regularity of embeddings of infinite-dimensional fractal sets into finitedimensional spaces, Nonlinearity 12 (1999) 1263-1275) and Robinson (Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity 22 (2009) 711-728) for subsets of Hilbert and Banach spaces with finite box-counting dimension are asymptotically sharp. An analogous argument allows us to obtain a lower bound for the power of the logarithmic correction term in an embedding theorem proved by Olson and Robinson (Almost bi-Lipschitz embeddings and almost homogeneous sets, Trans. Amer. Math. Soc. 362 (1) (2010) 145-168) for subsets X of Hilbert spaces when X − X has finite Assouad dimension.

Almost bi-Lipschitz embeddings and almost homogeneous sets

“…(4) Is the exponent γ in Theorem 5.6 (the power of the slog term) in any way optimal? (Pinto de Moura & Robinson [19] show that one can do no better than γ > 1 2 in general.) (5) Can one bound the Assouad dimension of the attractors of dissipative PDEs (or preferably the set of differences of solutions lying on such attractors)?…”

Embedding properties of sets with finite box-counting dimension

“…As mentioned in the beginning of the section, we do not know if the dual thickness is always bounded above by the box-counting dimension. However, Pinto de Moura and Robinson [14] relied on this fact to prove that the dual thickness and box-counting dimension of these orthogonal sequences coincide for every p ∈ [1, ∞]. In the next lemma, we show that this upper bound is true in this particular case.…”

“…In the remainder of this paper, we concentrate on a particular subset of ℓ p , for p ∈ [1, ∞], and prove that some of the inequalities we know so far are sharp. These sets were first discussed by Ben Artzi et al [3] and have been used by Pinto De Moura & Robinson [14] as examples to show that the Hölder exponent of the inverses in Hunt and Kaloshin's Theorem 1.5 is asymptotically sharp. Take p ≥ 1 and let (α n ) ∞ n=1 be a decreasing sequence such that α n → 0.…”

“…Finally, in Section 4 we look closely at the thickness and dual thickness and how they relate to the box-counting dimension. We prove some general estimates and we also look at a particular class of sequences in ℓ p which was used by Pinto de Moura and Robinson [14] to prove that Theorem 1.5 is asymptotically sharp as k → ∞.…”

Embedding of global attractors and their dynamics

“…Pinto de Moura and Robinson (2010a) showed that this bound on the logarithmic exponent γ in Theorem 4.1 is sharp as m −→ ∞. 2 The term 'prevalence' was coined by Hunt et al (1992) and generalizes the notion of 'Lebesgue almost every' from finite to infinite-dimensional spaces.…”