arXiv subjects

Yaghoub Rahimi

Publications and source records attributed to Yaghoub Rahimi.

9 recordsLinked to original sources

arXiv2026

Primes with Restricted-Digit Differences

Fix a base $b\ge2$ and a digit set $\mathcal{D}\subset\{0,1,\dots,b-1\}$, and let $\mathcal C_k$ be the set of integers in $[0,b^k)$ whose $k$ base-$b$ digits all lie in $\mathcal D$. We study prime pairs and three-term arithmetic progressions in primes whose difference, or common difference, belongs to $\mathcal C_k$. Under a natural digit non-resonance condition, we establish a localized Fourier criterion guaranteeing asymptotic formulae with explicit leading constants for both problems. Using a transfer-operator argument, we verify this criterion in explicit parameter ranges for several natural families of digit sets.

math.NT↗

arXiv2026

On the small denominator problem for generalized Minkowski--Funk transforms

Rubin's generalized Minkowski--Funk transforms $M_t^\alpha$ on the sphere $\mathbb{S}^n$ give rise, for irrational radii $t=\cos(\beta\pi)$, to a small denominator problem governed by the asymptotic behavior of their spectral multipliers. We show that for Lebesgue-almost every $\beta$ the corresponding two-sine small divisor inequality has infinitely many solutions, and deduce that $(M_t^\alpha)^{-1}$ is not bounded from $\tilde{H}^{s+\rho+1}(\mathbb{S}^n)$ to $H^s(\mathbb{S}^n)$ in the non-critical case $\rho\neq 0,1$. In the critical cases $\rho\in\{0,1\}$ we prove Rubin's Conjectures 4.4 and 4.7 on the failure of endpoint Sobolev regularity for the inverse transforms.

math.CA↗

arXiv2024

A Density Theorem for Higher Order Sums of Prime Numbers

Let $P$ be a subset of the primes of lower density strictly larger than $\frac12$. Then, every sufficiently large even integer is a sum of four primes from the set $P$. We establish similar results for $k$-summands, with $k\geq 4$, and for $k \geq 4$ distinct subsets of primes. This extends the work of H.~Li, H.~Pan, as well as X.~Shao on sums of three primes, and A.~Alsteri and X.~Shao on sums of two primes. The primary new contributions come from elementary combinatorial lemmas.

math.NT↗

arXiv2023

Averages over the Gaussian Primes: Goldbach's Conjecture and Improving Estimates

We prove versions of Goldbach conjectures for Gaussian primes in arbitrary sectors. Fix an interval $\omega \subset \mathbb{T}$. There is an integer $N_\omega $, so that every odd integer $n$ with $N(n)>N_\omega $ and $\text{dist}( \text{arg}(n) , \mathbb{T}\setminus \omega ) > (\log N(n)) ^{-B}$, is a sum of three Gaussian primes $n=p_1+p_2+p_3$, with $\text{arg}(p_j) \in \omega $, for $j=1,2,3$. A density version of the binary Goldbach conjecture in a sector is also proved.

math.NT↗

arXiv2022

A Lifted $\ell_1 $ Framework for Sparse Recovery

Motivated by re-weighted $\ell_1$ approaches for sparse recovery, we propose a lifted $\ell_1$ (LL1) regularization which is a generalized form of several popular regularizations in the literature. By exploring such connections, we discover there are two types of lifting functions which can guarantee that the proposed approach is equivalent to the $\ell_0$ minimization. Computationally, we design an efficient algorithm via the alternating direction method of multiplier (ADMM) and establish the convergence for an unconstrained formulation. Experimental results are presented to demonstrate how this generalization improves sparse recovery over the state-of-the-art.

eess.SP↗

arXiv2022

Improving and Maximal Inequalities for Primes in Progressions

Assume that $ y < N$ are integers, and that $ (b,y) =1$. Define an average along the primes in a progression of diameter $ y$, given by integer $ (b,y)=1 $. \begin{align*} A_{N,y,b} := \frac{ϕ(y)}{N} \sum _{\substack{n N _{y,r}} \lvert A_{N,y,b} f \rvert \rVert_{r}\ll \lVert f\rVert_{r}. \end{align*} The implied constant is only a function of $ r$. The uniformity over progressions imposes several novel elements on the proof.

math.CA↗

arXiv2021

Endpoint $ \ell ^{r}$ improving estimates for Prime averages

Let $ \Lambda $ denote von Mangoldt's function, and consider the averages \begin{align*} A_N f (x) &=\frac{1}{N}\sum_{1\leq n \leq N}f(x-n)\Lambda(n) . \end{align*} We prove sharp $ \ell ^{p}$-improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets $ F, G\subset [0,N]$ there holds \begin{equation*} N ^{-1} \langle A_N \mathbf 1_{F} , \mathbf 1_{G} \rangle \ll \frac{\lvert F\rvert \cdot \lvert G\rvert} { N ^2 } \Bigl( \operatorname {Log} \frac{\lvert F\rvert \cdot \lvert G\rvert} { N ^2 } \Bigr) ^{t}, \end{equation*} where $ t=2$, or assuming the Generalized Riemann Hypothesis, $ t=1$. The corresponding sparse bound is proved for the maximal function $ \sup_N A_N \mathbf 1_{F}$. The inequalities for $ t=1$ are sharp. The proof depends upon the Circle Method, and an interpolation argument of Bourgain.

math.NT↗

arXiv2020

Accelerated Schemes for the $L_1/L_2$ Minimization

In this paper, we consider the $L_1/L_2 $ minimization for sparse recovery and study its relationship with the $L_1$-$ αL_2 $ model. Based on this relationship, we propose three numerical algorithms to minimize this ratio model, two of which work as adaptive schemes and greatly reduce the computation time. Focusing on two adaptive schemes, we discuss their connection to existing approaches and analyze their convergence. The experimental results demonstrate the proposed approaches are comparable to the state-of-the-art methods in sparse recovery and work particularly well when the ground-truth signal has a high dynamic range. Lastly, we reveal some empirical evidence on the exact $L_1$ recovery under various combinations of sparsity, coherence, and dynamic ranges, which calls for theoretical justification in the future.

math.NA↗

arXiv2019

A Scale Invariant Approach for Sparse Signal Recovery

In this paper, we study the ratio of the $L_1 $ and $L_2 $ norms, denoted as $L_1/L_2$, to promote sparsity. Due to the non-convexity and non-linearity, there has been little attention to this scale-invariant model. Compared to popular models in the literature such as the $L_p$ model for $p\in(0,1)$ and the transformed $L_1$ (TL1), this ratio model is parameter free. Theoretically, we present a strong null space property (sNSP) and prove that any sparse vector is a local minimizer of the $L_1 /L_2 $ model provided with this sNSP condition. Computationally, we focus on a constrained formulation that can be solved via the alternating direction method of multipliers (ADMM). Experiments show that the proposed approach is comparable to the state-of-the-art methods in sparse recovery. In addition, a variant of the $L_1/L_2$ model to apply on the gradient is also discussed with a proof-of-concept example of the MRI reconstruction.

math.NA↗