Jack Brown (August 14, 1938- December 10, 2022) had a long, a varied, and a productive career that had an extraordinarily positive influence on several generations of students, the mathematical community of scholars, and the national security of the United States.

Jack Brown (August 14, 1938 -- December 10, 2022). Three former Ph.D. students of Jack Brown reflect on the magnificent ways that he challenged them to think and shaped their careers through his careful guiding touch.

We survey our recent result that for every continuous function there is an absolutely continuous homeomorphism such that the composition has a uniformly converging Fourier expansion. We mention the history of the problem, orginally stated by Luzin, and some details of the proof.

A conjecture of Erdös states that for any infinite set $A \subseteq \mathbb R$, there exists a Borel set $E \subseteq \mathbb R$ of positive Lebesgue measure that does not contain any non-trivial affine copy of $A$. The conjecture remains open for most fast-decaying sequences, including the geometric sequence $A = \{2^{-k} : k \geq 1\}$. In this article, we consider infinite decreasing sequences $A = \{a_k: k \geq 1\}$ in $\R$ that converge to zero at a prescribed rate; namely $\log (a_n/a_{n+1}) = e^{\varphi(n)} $, where $\varphi(n)/n\to 0$ as $n\to\infty$. This condition is satisfied by sequences whose logarithm has polynomial decay, and in particular by the geometric sequence. For any such sequence $A$, we construct a Cantor set $K \subseteq \mathbb [0,1]$ with measure arbitrarily close to 1, such that the set of Erdös points $\mathcal{E}\subseteq K$ has Hasudorff dimension 1.

A construction of product measures is given for an arbitrary sequence of measure spaces via outer measure techniques without imposing any condition on the underlying measure spaces. This result generalizes the ones given up to date.

Let $(X,\mathcal{F},\mu,T)$ be a not necessarily invertible non-atomic measure-preserving dynamical system where the $\sigma$-algebra $\mathcal{F}$ is generated by the intervals according to some total order. The main result is that the classical Rokhlin lemma may be adapted to such a situation assuming a slight extension of aperiodicity. This result is compared to previous noninvertible versions of the Rokhlin lemma.

The aim of this research paper is to establish some new generalized Gamidov type integral inequalities involving a $\psi $-fractional operator. We also give two applications to substantiate the validity of our findings.

We study a diffeomorphism which admits a weak hyperbolic product structure region, which is the intersection of two transversal families of weak stable and weak unstable disks, with countably many branches and integrable return times. We show that for such maps the distributions of the number of visits to a ball $B(x, r)$ converges to a Poisson distributions as the radius $r \to 0$. Applications of our resutls are some partially hyperbolic diffeomorphisms of which restriction on one dimensional unstable direction behaves as Manneville-Pomeau maps.

For a sequence in a Banach space $\mathcal{X}$, it is known that the set of subsequential limits of the sequence forms a closed subset of $\mathcal{X}$. Similarly, if the sequence is convergent, then the sequence of its Cesàro averages also converge to the same value. In this article, we study the properties of the set of Cesàro limits of subsequences of a given sequence in a Banach space using techniques from ergodic theory.

We study some variants of the Erdös similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset $E$ of the real line such that $0$ is a Lebesgue density point of $E$, but $E$ does not contain any (non-constant) infinite geometric progression. We give a sufficient density type condition that guarantees that a set contains an infinite geometric progression. By slightly improving a recent result of Bradford, Kohut and Mooroo-gen we construct a closed set $F\subset[0,\infty)$ such that the measure of $F\cap[t,t+1]$ tends to $1$ at infinity but $F$ does not contain any infinite arithmetic progression. We also slightly improve a more general recent result by Kolountzakis and Papageorgiou for more general sequences. We give a sufficient condition that guarantees that a given Cantor type set contains at least one infinite geometric progression with any quotient between $0$ and $1$. This can be applied to most symmetric Cantor sets of positive measure.

We survey the effective foundations for analysis with infinitesimals developed by Hrbacek and Katz in 2021, and detail some applications. Theories SPOT and SCOT are conservative over respectively ZF and ZF+ADC. The range of applications of these theories illustrates the fact that analysis with infinitesimals requires no more choice than traditional analysis. The theory SCOT incorporates in particular all the axioms of Nelson's Radically Elementary Probability Theory, which is therefore conservative over ZF+ADC.

We generalize Mitsis’s work to higher dimensions and give application to a conditional estimate concerning Omega-Kakeya sets. Also, following Bourgain's arithmetic argument, we improve concrete bound on the Hausdorff dimension of an Omega-Kakeya set.

We prove the existence of hypercyclic algebras for a class of convolution operators $\varphi(D)$ in which the value $\varphi(0)$ plays no role.

Motivated by recent studies on the notions of lineability and spaceability in the context of linear dynamics, we investigate the existence of infinite-dimensional closed subspaces of frequently hypercyclic vectors for frequently hypercyclic composition operators, known in the literature as Koopman operators and extensively used in many applications (like, for instance, the analysis of the dynamics of economic models formulated in terms of dynamical systems). All the results are obtained on $L^p$ spaces, $1 \leq p < \infty$, and in the dissipative setting with the extra hypothesis of bounded distortion. This allows us, as a consequence, to deduce analogous conclusions for fundamental mathematical objects: bilateral weighted backward shifts on $\ell^p$ spaces.