Sárközy's Theorem for Fractional Monomials
Suppose $A$ is a subset of $\{1, \dotsc, N\}$ which does not contain any configurations of the form $x,x+\lfloor n^c \rfloor$ where $n \neq 0$ and $1 0$.
math.NT↗
arXiv subjects
Maximilian O'Keeffe
Publications and source records attributed to Maximilian O'Keeffe.
Variational Estimates for Bilinear Ergodic Averages Along Sublinear Sequences
We prove long variational estimates for the bilinear ergodic averages \[ A_{N;X}(f,g)(x) = \frac{1}{N} \sum_{n=1}^N f(T^{\lfloor \sqrt{n} \rfloor}x) g(T^nx) \] on an arbitrary measure preserving system $(X,μ,T)$ for the full expected range, i.e. whenever $f \in L^{p_1}(X)$ and $g \in L^{p_2}(X)$ with $1 \frac{1}{2}$, which is sharp up to the endpoint. If $p \geq 1$ we obtain long variational estimates for the full expected range $r>2$ and if $p<1$ we obtain a range of $r>2+\varepsilon_{p_1,p_2}$ where $\varepsilon_{p_1,p_2}>0$ depends only on $p_1$ and $p_2$. As a consequence, we obtain bilinear maximal estimates \[ \left\| \sup_{N \in \mathbb{N}} |A_{N;X}(f,g)| \right\|_{L^p(X)} \leq C_{p_1,p_2} \|f\|_{L^{p_1}(X)} \|g\|_{L^{p_2}(X)} \] for any $1<p_1,p_2 \leq \infty$.
math.DS↗
Pointwise Convergence of Ergodic Averages Along Hardy Field Sequences
Let $(X,μ)$ be an arbitrary measure space equipped with a family of pairwise commuting measure preserving transformations $T_1, \dotsc, T_m$. We prove that the ergodic averages \[ A_{N;X}^{P_1, \dotsc, P_m}f = \frac{1}{N} \sum_{n=1}^N T_1^{\lfloor P_1(n) \rfloor} \dotsm T_m^{\lfloor P_m(n) \rfloor} f \] converge pointwise $μ$-almost everywhere as $N \to \infty$ for any $f \in L^p(X)$ with $p>1$, where $P_1, \dotsc, P_m$ are Hardy field functions which are "non-polynomial" and have distinct growth rates. To establish pointwise convergence we will prove a long-variational inequality, which will in turn prove that a maximal inequality holds for our averages. Additionally, by restricting the class of Hardy field functions to those with the same growth rate as $t^c$ for $c>0$ non-integer, we also prove full variational estimates. We are therefore able to provide quantitative bounds on the rate of convergence of exponential sums of the form \[ \frac{1}{N} \sum_{n=1}^N e(ξ_1 \lfloor n^{c_1} \rfloor + \dotsb + \lfloor n^{c_m} \rfloor) \] where $0<c_1<\dotsb<c_m$ are non-integer.
math.DS↗