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1. Richter’s local limit theorem, its refinement, and related results*

   https://doi.org/10.1007/s10986-023-09598-9  

   Bobkov, Sergey G.; Chistyakov, Gennadiy P.; Götze, Friedrich (April 2023, Lithuanian Mathematical Journal)

   We give a detailed exposition of the proof of Richter’s local limit theorem in a refined form and establish the stability of the remainder term in this theorem under small perturbations of the underlying distribution (including smoothing).We also discuss related quantitative bounds for characteristic functions and Laplace transforms. 

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2. Expected Number of Real Zeros for Random Orthogonal Polynomials

   Lubinsky, Doron; Pritsker, Igor; Xie, Xiaoyou (January 2018, Mathematical proceedings of the Cambridge Philosophical Society)

   We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only $$(2/\pi + o(1))\log{n}$$ expected real zeros in terms of the degree $$n$$. If the basis is given by the orthonormal polynomials associated with a compactly supported Borel measure on the real line, or associated with a Freud weight, then random linear combinations have $$n/\sqrt{3} + o(n)$$ expected real zeros. We prove that the same asymptotic relation holds for all random orthogonal polynomials on the real line associated with a large class of weights, and give local results on the expected number of real zeros. We also show that the counting measures of properly scaled zeros of these random polynomials converge weakly to either the Ullman distribution or the arcsine distribution. 

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3. Differential Privacy on Finite Computers

   https://doi.org/10.29012/jpc.679  

   Balcer, Victor; Vadhan, Salil (October 2019, Journal of Privacy and Confidentiality)

   We consider the problem of designing and analyzing differentially private algorithms that can be implemented on discrete models of computation in strict polynomial time, motivated by known attacks on floating point implementations of real-arithmetic differentially private algorithms (Mironov, CCS 2012) and the potential for timing attacks on expected polynomial-time algorithms. As a case study, we examine the basic problem of approximating the histogram of a categorical dataset over a possibly large data universe X. The classic Laplace Mechanism (Dwork, McSherry, Nissim, Smith, TCC 2006 and J. Privacy \& Confidentiality 2017) does not satisfy our requirements, as it is based on real arithmetic, and natural discrete analogues, such as the Geometric Mechanism (Ghosh, Roughgarden, Sundarajan, STOC 2009 and SICOMP 2012), take time at least linear in |X|, which can be exponential in the bit length of the input.   In this paper, we provide strict polynomial-time discrete algorithms for approximate histograms whose simultaneous accuracy (the maximum error over all bins) matches that of the Laplace Mechanism up to constant factors, while retaining the same (pure) differential privacy guarantee. One of our algorithms produces a sparse histogram as output. Its ``"per-bin accuracy" (the error on individual bins) is worse than that of the Laplace Mechanism by a factor of log|X|, but we prove a lower bound showing that this is necessary for any algorithm that produces a sparse histogram. A second algorithm avoids this lower bound, and matches the per-bin accuracy of the Laplace Mechanism, by producing a compact and efficiently computable representation of a dense histogram; it is based on an (n+1)-wise independent implementation of an appropriately clamped version of the Discrete Geometric Mechanism. 

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4. An unconditional Montgomery theorem for pair correlation of zeros of the Riemann zeta-function

   https://doi.org/10.4064/aa230612-20-3  

   Baluyot, Siegfred_Alan C; Goldston, Daniel Alan; Suriajaya, Ade Irma; Turnage-Butterbaugh, Caroline L (April 2024, Acta Arithmetica)

   Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem concerning pair correlation of zeros of the Riemann zeta-function. One consequence of this theorem is that, assuming RH, at least 67.9% of the nontrivial zeros are simple. Here we obtain an unconditional form of Montgomery’s theorem and show how to apply it to prove the following result on simple zeros: If all the zeros ρ=β+iγ of the Riemann zeta-function such that T3/8<γ≤T satisfy ∣∣β−1/2∣∣<1/(2logT), then, as T tends to infinity, at least 61.7% of these zeros are simple. The method of proof neither requires nor provides any information on whether any of these zeros are or are not on the critical line where β=1/2. We also obtain the same result under the weaker assumption of a strong zero-density hypothesis. 

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5. Rényi divergences in central limit theorems: Old and new

   https://doi.org/10.1214/25-PS30  

   Bobkov, Sergey G; Götze, Friedrich (January 2025, Probability Surveys)

   We give an overview of various results and methods related to information-theoretic distances of Rényi type in the light of their applications to the central limit theorem (CLT). The first part (Sections 1–9) is devoted to the total variation and the Kullback-Leibler distance (relative entropy). In the second part (Sections 10–15) we discuss general properties of Rényi and Tsall is divergences of order alpha > 1, and then in the third part (Sections 16–21) we turn to the CLT and non-uniform local limit theorems with respect to these strong distances. In the fourth part (Sections 22–31), we discuss recent results on strictly subgaussian distributions and describe necessary and sufficient conditions which ensure the validity of the CLT with respect to the Rényi divergence of infinite order. 

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