arXiv subjects

Anastasios Fragkos

Publications and source records attributed to Anastasios Fragkos.

11 recordsLinked to original sources

arXiv2026

Pointwise Ergodic Theorems Along $\lfloor p^c\rfloor$ for Every $c>1$

Let $c>1$ be noninteger, and let $Λ$ denote the von Mangoldt function. The averages \[ A_N^{c} f(x):=\frac1N\sum_{n=1}^{N} Λ(n)f\bigl(T^{\lfloor n^c\rfloor}x\bigr) \] satisfy the variational estimate \[ \bigl\|\mathcal{V}^r(A_N^{c} f:N\in\mathbb{N})\bigr\|_{L^p(X)} \lesssim_{c,p,r}\|f\|_{L^p(X)} \] for every $1 2$, on every $σ$-finite measure-preserving system. In particular, the normalized averages along fractional powers of primes converge almost everywhere. The main arithmetic ingredient is a uniform bound on the correlations of $Λ-1$ with fractional-power exponential phases, yielding arbitrarily large powers of logarithmic gain. We obtain this estimate through short-interval polynomial approximation of the fractional phase and higher-order discorrelation estimates. The same approach yields almost-everywhere convergence to for the Möbius-weighted averages along $\left( \lfloor n^c \rfloor \right)_{n \in \mathbb{N}}$ throughout $1<p<\infty$.

math.CA↗

arXiv2026

Reasonable bounds for Restricted Polynomial Progressions in $\mathbb F^n_p$

Given an integer $k \geq 2$, there are integers $M_k$ and $P_k$ so that for all $A \subseteq \mathbb F^n_p$ does not contain a restricted polynomial progression of the form \begin{align*} x, x+\begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}, x+\begin{pmatrix} y_1^2 \\ y_2^2 \\ \vdots \\ y_n^2 \end{pmatrix}, \ldots, x+\begin{pmatrix} y_1^{k-1} \\ y_2^{k-1} \\ \vdots \\ y_n^{k-1} \end{pmatrix}, \end{align*} where $x \in \mathbb F^n_p$, with $p>P_k$, and $y \in \{0,2, 3, \ldots, M_k\}^n \backslash \{\vec{0} \}$, for some fixed $M_k$, then $A$ must have density bounded above by $\frac{C}{(\log^{(k^{O(k)})} n)^c}$, where $C, c>0$ only depend on $k$ and $p$. For $k\geq 4$, the best previously known bound came from the density Hales--Jewett theorem and was of inverse Ackermann type. The proof uses a recent inverse theorem for noise stability of Bhangale, Khot, Liu and Minzer, which arose in the study of constraint satisfaction problems. The main new ingredient is a proof that the underlying distribution counting restricted polynomial progressions does not admit a nontrivial abelian embedding, by way of an involved algebraic argument. Reasonable bounds are provided for $P_k$ and $M_k$.

math.NT↗

arXiv2026

Szemerédi's Theorem Along Cantor Sets of Integers

Let $\mathcal C= \{k_1 0. $$ The averages above are uniform in the index $n$, and so the set of indices $n$ for which $m(A\cap T^{-k_n} A \cap \cdots \cap T^{-\ell k_n} A )>0$ is syndetic. By the Furstenberg correspondence principle, it follows that for every set of integers $A$ of positive upper Banach density there is a syndetic set $B$ of indices such that, for each $n \in B$, the set of $a$ with $a, a+k_n, \ldots, a+ \ell k_n \in A$ again has positive upper Banach density. A Cantor set contains an IP-set, so the existence of such progressions, and the positivity of the density above for a single index, already follow from the IP Szemerédi theorem of Furstenberg and Katznelson, and from the work of Kra and Shalom. The point of this paper is instead the frequency with which the step sizes $k_n$ may be found, measured in the natural enumeration of the Cantor set itself.

math.NT↗

arXiv2026

Integer Cantor Sets: Arithmetic Combinatorial Properties

Cantor sets of integers have a rich set of arithmetic combinatorial properties. We consider classical Cantor sets, with a base and a fixed set of allowed digits. For such sets, we (a) give examples of such sets that satisfy the intersective property with power savings (b) characterize uniform distribution, (c) establish polynomial mean ergodic theorems and (d) study metric pair correlation of Cantor sets.

math.DS↗

arXiv2026

Endpoint Estimates for Stein's Purely Quadratic Carleson Operator

We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform \[ \mathcal{C}_2f(x) := \sup_λ \left|\operatorname{p.v.} \int_{\mathbb{R}}f(x-y)e^{2 πi λy^2} \frac{\mathrm{d} y }{y} \right| \] and its lacunary counterpart obtained by restricting $λ$ to $2^{\mathbb{Z}}.$ We prove that if $Φ$ is a Young function satisfying $ Φ(t)=o\bigl(t\log_2t\bigr)$ then $\mathcal{C}_{2,\mathsf{lac}}$, and therefore $\mathcal{C}_2$ as well, does not satisfy a corresponding $Φ$-modular estimate. In particular, neither the lacunary nor the full quadratic operator is of weak type $(1,1)$. In the positive direction, we establish an $L\log_1 L$ modular estimate for $\mathcal{C}_2$ and a $L (\log_2L)^2 \log_4L$ estimate for $\mathcal{C}_{2,\mathsf{lac}}.$

math.CA↗

arXiv2026

Polynomial Ergodic Averages Along Short Intervals

We study pointwise convergence of polynomial ergodic averages over short intervals whose left endpoints tend to infinity. For a polynomial orbit of degree $d\geq2$ and doubly lacunary starting times, we prove $L^p$ variational estimates, and hence almost-everywhere convergence, for $1 (d-1)/d$. This gives the first pointwise ergodic theorem for polynomial orbits along short intervals. We also show that the endpoint $L^1$ fails along every infinite subsequence. In a different direction, we prove that substantially denser sequences of starting times exhibit the strong sweeping-out property.

math.CA↗

arXiv2026

The Wiener Wintner Theorem Along the Primes

We prove the following Wiener-Wintner Theorem along the sequence of prime times, the first extension of the Wiener-Wintner Theorem to arithmetic sequences: for every probability space, $(X, ν),$ equipped with a measure-preserving transformation, $T : X \to X,$ and every $f \in L^p(X), 1 < p \leq \infty$, there exists a set of full probability, $X_f \subset X$ with $ν(X_f) = 1,$ so that for all $ω\in X_f$, \[ \frac{1}{N} \sum_{n \leq N} e^{ 2 πi p_n θ} f(T^{p_n} ω) \] converges for all $θ\in [0,1]$; above, $\{2 = p_1 < p_2 < \dots\}$ are an enumeration of the primes. Our proof lives at the interface of classical Fourier analysis, combinatorial number theory, higher order Fourier analysis, and pointwise ergodic theory, with U^3 theory playing an important role; our $U^3$-estimates for Heath-Brown models of the von Mangoldt function may be of independent interest.

math.DS↗

arXiv2026

A Variation Norm Carleson Theorem Along the Primes

Let $Λ$ denote the von Mangoldt function; we prove that for each $r > 2$, there exist constants \[ r' < \mathbf{c}(r) < 2 < \mathbf{C}(r), \qquad \lim_{r \to \infty} \mathbf{c}(r) = 1, \ \lim_{r \to \infty} \mathbf{C}(r) = \infty \] so that the discrete variational Carleson operator along the primes \begin{align} \mathcal{V}^r \Big( \sum_{n \neq 0} f(x-n) Λ(|n|) \frac{e^{2πi λn}}{n} : λ\in \mathbb{T} \Big) \end{align} is bounded on $\ell^p$ for all $\mathbf{c}(r) < p < \mathbf{C}(r)$, while the variation is unbounded when $p \leq r'$. At the non-variational endpoint, the same argument gives the sharp maximal result: the prime Carleson operator \[ \sup_{λ\in\mathbb T} \Big|\sum_{n\neq0} f(x-n)Λ(|n|)\frac{e^{2πiλn}}{n}\Big| \] is bounded on $\ell^p(\mathbb Z)$ for the full expected range $1<p<\infty$. The proof gives a new mechanism for treating modulation-invariant singular integrals after arithmetic sparsification. It combines higher-order Fourier uniformity, a variable-coefficient multi-frequency principle in the spirit of Bourgain, and an additive-combinatorial inverse argument. A key step is a reduction to finite periodic models, where the Ramanujan structure of the major arcs is converted into a sharp estimate for structured atoms by elementary number theory.

math.CA↗

arXiv2026

Fractionally modulated discrete Carleson's Theorem and pointwise Ergodic Theorems along certain curves

For $c\in(1,2)$ we consider the following operators \[ \mathcal{C}_{c}f(x) = \sup_{λ\in [-1/2,1/2)}\bigg| \sum_{n \neq 0}f(x-n) \frac{e^{2πiλ\lfloor |n|^{c} \rfloor}}{n}\bigg|\text{,}\quad \mathcal{C}^{\mathsf{sgn}}_{c}f(x) = \sup_{λ\in [-1/2,1/2)}\bigg| \sum_{n \neq 0}f(x-n) \frac{e^{2πiλ\mathsf{sign(n)} \lfloor |n|^{c} \rfloor}}{n}\bigg| \text{,} \] and prove that both extend boundedly on $\ell^p(\mathbb{Z})$, $p\in(1,\infty)$. The second main result is establishing almost everywhere pointwise convergence for the following ergodic averages \[ A_Nf(x)=\frac{1}{N}\sum_{n=1}^Nf(T^nS^{\lfloor n^c\rfloor}x)\text{,} \] where $T,S\colon X\to X$ are commuting measure-preserving transformations on a $σ$-finite measure space $(X,μ)$, and $f\in L_μ^p(X)$, $p\in(1,\infty)$. The point of departure for both proofs is the study of exponential sums with phases $ξ_2 \lfloor |n^c|\rfloor+ ξ_1n$ through the use of a simple variant of the circle method.

math.DS↗

arXiv2025

Multilinear Wavelet Compact T(1) Theorem

We prove a wavelet $T(1)$ theorem for compactness of multilinear Calderón-Zygmund (CZ) operators. Our approach characterizes compactness in terms of testing conditions and yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator.

math.CA↗

arXiv2022

The weak-type Carleson theorem via wave packet estimates

We prove that the weak-$L^{p}$ norms, and in fact the sparse $(p,1)$-norms, of the Carleson maximal partial Fourier sum operator are $\lesssim (p-1)^{-1}$ as $p\to 1^+$. This is an improvement on the Carleson-Hunt theorem, where the same upper bound on the growth order is obtained for the restricted weak-$L^p$ type norm, and which was the strongest quantitative bound prior to our result. Furthermore, our sparse $(p,1)$-norms bound imply new and stronger results at the endpoint $p=1$. In particular, we obtain that the Fourier series of functions from the weighted Arias de Reyna space $ QA_{\infty}(w) $, which contains the weighted Antonov space $L\log L\log\log\log L(\mathbb T; w)$, converge almost everywhere whenever $w\in A_1$. This is an extension of the results of Antonov and Arias De Reyna, where $w$ must be Lebesgue measure. The backbone of our treatment is a new, sharply quantified near-$L^1$ Carleson embedding theorem for the modulation-invariant wave packet transform. The proof of the Carleson embedding relies on a newly developed smooth multi-frequency decomposition which, near the endpoint $p=1$, outperforms the abstract Hilbert space approach of past works, including the seminal one by Nazarov, Oberlin and Thiele. As a further example of application, we obtain a quantified version of the family of sparse bounds for the bilinear Hilbert transforms due to Culiuc, Ou and the first author.

math.CA↗