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See detailQuantitative CLTs on the Poisson space via Skorohod estimates and p-Poincaré inequalities
Trauthwein, Tara UL

in arXiv preprint (2022)

We establish new explicit bounds on the Gaussian approximation of Poisson functionals based on novel estimates of moments of Skorohod integrals. Combining these with the Malliavin-Stein method, we derive ... [more ▼]

We establish new explicit bounds on the Gaussian approximation of Poisson functionals based on novel estimates of moments of Skorohod integrals. Combining these with the Malliavin-Stein method, we derive bounds in the Wasserstein and Kolmogorov distances whose application requires minimal moment assumptions on add-one cost operators – thereby extending the results from (Last, Peccati and Schulte, 2016). Our applications include a CLT for the Online Nearest Neighbour graph, whose validity was conjectured in (Wade, 2009; Penrose and Wade, 2009). We also apply our techniques to derive quantitative CLTs for edge functionals of the Gilbert graph, of the k-Nearest Neighbour graph and of the Radial Spanning Tree, both in cases where qualitative CLTs are known and unknown. [less ▲]

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See detailThe distribution of rational numbers on Cantor's middle thirds set
Trauthwein, Tara UL; Rahm, Alexander; Solomon, Noam et al

in Uniform distribution theory (2020), 15(2), 73-92

We give a heuristic argument predicting that the number N*(T) of rationals p/q on Cantor’s middle thirds set C such that gcd(p, q) = 1 and q ≤ T, has asymptotic growth O(T^{d+ε}), for d = dim C. Our ... [more ▼]

We give a heuristic argument predicting that the number N*(T) of rationals p/q on Cantor’s middle thirds set C such that gcd(p, q) = 1 and q ≤ T, has asymptotic growth O(T^{d+ε}), for d = dim C. Our heuristic is related to similar heuristics and conjectures proposed by Fishman and Simmons. We also describe extensive numerical computations supporting this heuristic. Our heuristic predicts a similar asymptotic if C is replaced with any similar fractal with a description in terms of missing digits in a base expansion. Interest in the growth of N*(T) is motivated by a problem of Mahler on intrinsic Diophantine approximation on C. [less ▲]

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