Let $f\left(x\right)\sim \Sigma a_n{e}^{2\pi inx},f*\left(x\right)\sim {\sum }_{n=0}^{\infty }a{*}_n\cos 2\pi nx$, where the $a{*}_n$ are the numbers $\left|a_n\right|$ rearranged so that $a_n^{*}\searrow 0$. Then for any convex increasing $\psi$, $\parallel \psi \left(\right|f{|}^2{\parallel }_1\le \parallel \psi \left(20\right|f*{|}^2{\parallel }_1$. The special case $\psi \left(t\right)=t^{q/2}$, $q\ge 2$, gives $\parallel f{\parallel }_q\le 5\parallel f*{\parallel }_q$ an equivalent of Littlewood.

Various techniques are presented for constructing $\Lambda$ (p) sets which are not $\Lambda (p+ϵ)$ for all $ϵ\>0$. The main result is that there is a $\Lambda$ (4) set in the dual of any compact abelian group which is not $\Lambda (4+ϵ)$ for all $ϵ\>0$. Along the way to proving this, new constructions are given in dual groups in which constructions were already known of $\Lambda$ (p) not $\Lambda (p+ϵ)$ sets, for certain values of $p$. The main new constructions in specific dual groups are: – there is a $\Lambda$ (2k) set which is not $\Lambda (2k+ϵ)$ in $\mathbf{Z}\left(2\right)\oplus \mathbf{Z}\left(2\right)\oplus \cdots$ for all $2\le k$, $k\in \mathbf{N}$ and...

Let $T:x\mapsto x+g$ be an ergodic translation on the compact group $C$ and $M\subseteq C$ a continuity set, i.e. a subset with topological boundary of Haar measure 0. An infinite binary sequence $\mathbf{a}:\mathbb{Z}\mapsto \{0,1\}$ defined by $\mathbf{a}\left(k\right)=1$ if $T^k\left(0_C\right)\in M$ and $\mathbf{a}\left(k\right)=0$ otherwise, is called a Hartman sequence. This paper studies the growth rate of $P_{\mathbf{a}}\left(n\right)$, where $P_{\mathbf{a}}\left(n\right)$ denotes the number of binary words of length $n\in \mathbb{N}$ occurring in $\mathbf{a}$. The growth rate is always subexponential and this result is optimal. If $T$ is an ergodic translation $x\mapsto x+\alpha$ $(\alpha =({\alpha }_1,...,{\alpha }_s))$ on ${𝕋}^s$ and $M$ is a box with...

Consider the difference equation $$\Delta x\left(n\right)+\sum _{i=1}^m{p}_i\left(n\right)x\left({\tau }_i\left(n\right)\right)=0,n\ge 0\left[\nabla x\left(n\right)-\sum _{i=1}^m{p}_i\left(n\right)x\left({\sigma }_i\left(n\right)\right)=0,n\ge 1\right],$$ where $\left(p_i\left(n\right)\right)$, $1\le i\le m$ are sequences of nonnegative real numbers, ${\tau }_i\left(n\right)$ [${\sigma }_i\left(n\right)$], $1\le i\le m$ are general retarded (advanced) arguments and $\Delta$ [$\nabla$] denotes the forward (backward) difference operator $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ [$\nabla x\left(n\right)=x\left(n\right)-x(n-1)$]. New oscillation criteria are established when the well-known oscillation conditions $$\underset{n\to \infty }{lim\; sup}\sum _{i=1}^m\sum _{j=\tau \left(n\right)}^n{p}_i\left(j\right)>1\left[\underset{n\to \infty }{lim\; sup}\sum _{i=1}^m\sum _{j=n}^{\sigma \left(n\right)}{p}_i\left(j\right)>1\right]$$ and $$\underset{n\to \infty }{lim\; inf}\sum _{i=1}^m\sum _{j={\tau }_i\left(n\right)}^{n-1}{p}_i\left(j\right)>\frac{1}{\mathrm{e}}\left[\underset{n\to \infty }{lim\; inf}\sum _{i=1}^m\sum _{j=n+1}^{{\sigma }_i\left(n\right)}{p}_i\left(j\right)>\frac{1}{\mathrm{e}}\right]$$ are not satisfied. Here $\tau \left(n\right)={max}_{1\le i\le m}{\tau }_i\left(n\right)$ $[\sigma \left(n\right)={min}_{1\le i\le m}{\sigma }_i\left(n\right)]$. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.