Related papers: The mixed Littlewood conjecture for pseudo-absolut…

Let p be a prime, and let f : Z/pZ -> R be a function with average value 0 and ||f||_A <= 1, where ||f||_A denotes the algebra norm (L^1 norm of the Fourier transform). Then f(x) is small for some x, specifically min_x |f(x)| is no more…

The paper substantiates the conjecture of the asymptotic behavior of the largest distance between consecutive primes: $sup_{p_i \leq x}(p_{i+1}-p_i) \sim 2e^{-\gamma} \log^2(x)$, where $\gamma$ is the Euler constant. The Hardy-Littlewood…

For any prime $p$ and real number and $\alpha$, the $p$-adic Littlewood Conjecture due to de Mathan and Teuli\'e asserts that \[\inf_{|m|\ge1}|m|_p\cdot |m|\cdot |\left\langle\alpha m\right\rangle|=0.\] Above, $|m|$ is the usual absolute…

The existence of unimodular forms with small norms on sequence spaces is crucial in a variety of problems in modern analysis. We prove that the infimum of $\left\Vert A\right\Vert $ over all unimodular $d$-linear (complex or real) forms $A$…

Using a sieve-theoretic argument, we show that almost all gaps $(p_n, p_{n+1})$ between consecutive primes $p_n, p_{n+1}$ contain a natural number $m$ whose least prime factor $p(m)$ is at least the length $p_{n+1} - p_n$ of the gap,…

We prove that analogues of the Hardy-Littlewood generalised twin prime conjecture for almost primes hold on average. Our main theorem establishes an asymptotic formula for the number of integers $n=p_1p_2 \leq X$ such that $n+h$ is a…

Let $\langle x\rangle$ denote the distance from $x\in\mathbb{R}$ to the set of integers $\mathbb{Z}$. The Littlewood Conjecture states that for all pairs $(\alpha,\beta)\in\mathbb{R}^{2}$ the product $q\langle q\alpha\rangle\langle…

The Littlewood conjecture, proven by Konyagin and McGehee-Pigno-Smith in the 1980s, states that if $A\subset \mathbb{Z}$ is a finite set of integers with $\lvert A\rvert=N$ then $\| \widehat{1_A}\|_1\geq c\log N$ for some absolute constant…

Assuming a $q$-variant of the prime $k$-tuple conjecture uniformly, we compute mixed moments of the number of primes in disjoint short intervals and progressions, respectively. This involves estimating the mean of singular series along…

A Littlewood polynomial is a polynomial of the form \[ f_n(x)=\sum_{k=0}^n \varepsilon_k x^k \] with $\varepsilon_k\in\{-1, 1\}$. Let $(\varepsilon_k)_{k \ge 0}$ be i.i.d. Rademacher coefficients. We show that the lower envelope of…

In the late 1980s, A. Premet conjectured that the variety of nilpotent elements of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic $p>0$ is irreducible. This conjecture remains open, but it…

The article presents a generalization of the classical Hardy-Littlewood conjecture concerning the density of prime tuples to the case of tuples consisting of almost-prime numbers (numbers with a specified quantity of prime divisors). The…

Given a compact metric space (X,d) equipped with a non-atomic, probability measure m and a real, positive decreasing function p we consider a `natural' class of limsup subsets La(p) of X. The classical limsup sets of `well approximable'…

For statistical decision problems with finite parameter space, it is well-known that the upper value (minimax value) agrees with the lower value (maximin value). Only under a generalized notion of prior does such an equivalence carry over…

The paper investigates various $p$-adic versions of Littlewood's conjecture, generalizing a set-up considered recently by de Mathan and Teulie. In many cases it is shown that the sets of exceptions to these conjectures have Hausdorff…

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…

Let $L$ be a convex cone of real random variables on the probability space $(\Omega,\mathcal{A},P_0)$. The existence of a probability $P$ on $\mathcal{A}$ such that $$ P \sim P_0,\quad E_P \abs{X}< \infty\, \text{ and } \, E_P(X) \leq 0\,…

The twin prime conjecture asserts that there are infinitely many pairs of primes that differ by two. While recent advances have improved our understanding of bounded prime gaps, the conjecture remains unresolved. This paper refines the…

We show that there exist absolute constants $\Delta > \delta > 0$ such that, for all $n \geqslant 2$, there exists a polynomial $P$ of degree $n$, with $\pm 1$ coefficients, such that $$\delta\sqrt{n} \leqslant |P(z)| \leqslant…

In this paper, we prove the twin prime conjecture showing that \begin{align} \sum \limits_{\substack{p\leq x\\p,p+2\in \mathbb{P}}}1\geq (1+o(1))\frac{x}{2\mathcal{C}\log^2 x}\nonumber \end{align} where $\mathcal{C}:=\mathcal{C}(2)>0$ fixed…