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1. A Zero-One Law for Uniform Diophantine Approximation in Euclidean Norm

   https://doi.org/10.1093/imrn/rnaa256  

   Kleinbock, Dmitry; Rao, Anurag (October 2020, International Mathematics Research Notices)

   Abstract We study a norm-sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet’s theorem. Its supremum norm case was recently considered by the 1st-named author and Wadleigh [ 17], and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of the suitably scaled norm ball. We use methods from geometry of numbers to generalize a result due to Andersen and Duke [ 1] on measure zero and uncountability of the set of numbers (in some cases, matrices) for which Minkowski approximation theorem can be improved. The choice of the Euclidean norm on $$\mathbb{R}^2$$ corresponds to studying geodesics on a hyperbolic surface, which visit a decreasing family of balls. An application of the dynamical Borel–Cantelli lemma of Maucourant [ 25] produces, given an approximation function $$\psi $$, a zero-one law for the set of $$\alpha \in \mathbb{R}$$ such that for all large enough $$t$$ the inequality $$\left (\frac{\alpha q -p}{\psi (t)}\right )^2 + \left (\frac{q}{t}\right )^2 < \frac{2}{\sqrt{3}}$$ has non-trivial integer solutions. 

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2. Double Spike Dirichlet Priors for Structured Weighting

   Lin, Huiming; Li, Meng (September 2022, Journal of machine learning research)

   Assigning weights to a large pool of objects is a fundamental task in a wide variety of applications. In this article, we introduce the concept of structured high-dimensional probability simplexes, in which most components are zero or near zero and the remaining ones are close to each other. Such structure is well motivated by (i) high-dimensional weights that are common in modern applications, and (ii) ubiquitous examples in which equal weights -- despite their simplicity -- often achieve favorable or even state-of-the-art predictive performance. This particular structure, however, presents unique challenges partly because, unlike high-dimensional linear regression, the parameter space is a simplex and pattern switching between partial constancy and sparsity is unknown. To address these challenges, we propose a new class of double spike Dirichlet priors to shrink a probability simplex to one with the desired structure. When applied to ensemble learning, such priors lead to a Bayesian method for structured high-dimensional ensembles that is useful for forecast combination and improving random forests, while enabling uncertainty quantification. We design efficient Markov chain Monte Carlo algorithms for implementation. Posterior contraction rates are established to study large sample behaviors of the posterior distribution. We demonstrate the wide applicability and competitive performance of the proposed methods through simulations and two real data applications using the European Central Bank Survey of Professional Forecasters data set and a data set from the UC Irvine Machine Learning Repository (UCI). 

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3. The set of bounded continuous nowhere locally uniformly continuous functions is not Borel

   Alexander J. Izzo (January 2022, Houston journal of mathematics)

   It is known that for $$X$$ a nowhere locally compact metric space, the set of bounded continuous, nowhere locally uniformly continuous real-valued functions on $$X$$ contains a dense $$G_\delta$$ set in the space $$C_b(X)$$ of all bounded continuous real-valued functions on $$X$$ in the supremum norm. Furthermore, when $$X$$ is separable, the set of bounded continuous, nowhere locally uniformly continuous real-valued functions on $$X$$ is itself a $$G_\delta$$ set. We show that in contrast, when $$X$$ is nonseparable, this set of functions is not even a Borel set. 

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4. Normal measures on large cardinals

   https://doi.org/10.1090/btran/132  

   Apter, Arthur; Cummings, James (February 2023, Transactions of the American Mathematical Society, Series B)

   The space of normal measures on a measurable cardinal is naturally ordered by the Mitchell ordering. In the first part of this paper we show that the Mitchell ordering can be linear on a strong cardinal where the Generalised Continuum Hypothesis fails. In the second part we show that a supercompact cardinal at which the Generalised Continuum Hypothesis fails may carry a very large number of normal measures of Mitchell order zero. 

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5. Equal sums in random sets and the concentration of divisors

   https://doi.org/10.1007/s00222-022-01177-y  

   Ford, Kevin; Green, Ben; Koukoulopoulos, Dimitris (June 2023, Inventiones mathematicae)

   Abstract We study the extent to which divisors of a typical integer n are concentrated. In particular, defining $$\Delta (n) := \max _t \# \{d | n, \log d \in [t,t+1]\}$$ Δ ( n ) : = max t # { d | n , log d ∈ [ t , t + 1 ] } , we show that $$\Delta (n) \geqslant (\log \log n)^{0.35332277\ldots }$$ Δ ( n ) ⩾ ( log log n ) 0.35332277 … for almost all n , a bound we believe to be sharp. This disproves a conjecture of Maier and Tenenbaum. We also prove analogs for the concentration of divisors of a random permutation and of a random polynomial over a finite field. Most of the paper is devoted to a study of the following much more combinatorial problem of independent interest. Pick a random set $${\textbf{A}} \subset {\mathbb {N}}$$ A ⊂ N by selecting i to lie in $${\textbf{A}}$$ A with probability 1/ i . What is the supremum of all exponents $$\beta _k$$ β k such that, almost surely as $$D \rightarrow \infty $$ D → ∞ , some integer is the sum of elements of $${\textbf{A}} \cap [D^{\beta _k}, D]$$ A ∩ [ D β k , D ] in k different ways? We characterise $$\beta _k$$ β k as the solution to a certain optimisation problem over measures on the discrete cube $$\{0,1\}^k$$ { 0 , 1 } k , and obtain lower bounds for $$\beta _k$$ β k which we believe to be asymptotically sharp. 

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