scholarly journals Perfectly ordered quasicrystals and the Littlewood conjecture

2018 ◽  
Vol 370 (7) ◽  
pp. 4975-4992
Author(s):  
Alan Haynes ◽  
Henna Koivusalo ◽  
James Walton
Keyword(s):  
Littlewood Conjecture

The Discrepancy of (nkx)k=1∞ With Respect to Certain Probability Measures

2020 ◽  
Vol 71 (2) ◽  
pp. 573-597
Author(s):  
Niclas Technau ◽  
Agamemnon Zafeiropoulos
Keyword(s):  
Probability Measure ◽  
Lacunary Sequence ◽  
Probability Measures ◽  
Littlewood Conjecture ◽  
Almost All

Abstract Let $(n_k)_{k=1}^{\infty }$ be a lacunary sequence of integers. We show that if $\mu$ is a probability measure on $[0,1)$ such that $|\widehat{\mu }(t)|\leq c|t|^{-\eta }$, then for $\mu$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies $$\begin{equation*}\frac{1}{4} \leq \limsup_{N\to\infty}\frac{N D_N(n_kx)}{\sqrt{N\log\log N}} \leq C\end{equation*}$$for some constant $C>0$. This proves a conjecture of Haynes, Jensen and Kristensen and allows an improvement on their previous result relevant to an inhomogeneous version of the Littlewood conjecture.

Hardy Inequalities on the Real Line

2011 ◽  
Vol 54 (1) ◽  
pp. 159-171 ◽  
Author(s):  
Mohammad Sababheh
Keyword(s):  
Real Line ◽  
Hardy’S Inequality ◽  
Hardy Inequalities ◽  
The Real ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Hardy's Inequality ◽  
Littlewood Conjecture ◽  
The Relationship

AbstractWe prove that some inequalities, which are considered to be generalizations of Hardy's inequality on the circle, can be modified and proved to be true for functions integrable on the real line. In fact we would like to show that some constructions that were used to prove the Littlewood conjecture can be used similarly to produce real Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type inequalities on the circle and those on the real line.

scholarly journals METRIC CONSIDERATIONS CONCERNING THE MIXED LITTLEWOOD CONJECTURE

2011 ◽  
Vol 07 (03) ◽  
pp. 593-609 ◽  
Author(s):  
YANN BUGEAUD ◽  
ALAN HAYNES ◽  
SANJU VELANI
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Metrical Theory ◽  
Littlewood Conjecture

The main goal of this paper is to develop a metrical theory of Diophantine approximation within the framework of the de Mathan–Teulié Conjecture — also known as the "Mixed Littlewood Conjecture". Let p be a prime. A consequence of our main result is that, for almost every real number α, [Formula: see text]

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