We use concepts of continuous higher randomness, developed in Bienvenu et al. [‘Continuous higher randomness’, J. Math. Log. 17(1) (2017).], to investigate $\unicode[STIX]{x1D6F1}_{1}^{1}$-randomness. We discuss lowness for $\unicode[STIX]{x1D6F1}_{1}^{1}$-randomness, cupping with $\unicode[STIX]{x1D6F1}_{1}^{1}$-random sequences, and an analogue of the Hirschfeldt–Miller characterization of weak 2-randomness. We also consider analogous questions for Cohen forcing, concentrating on the class of $\unicode[STIX]{x1D6F4}_{1}^{1}$-generic reals.

We prove the boundedness of a smooth bilinear Rubio de Francia operator associated with an arbitrary collection of squares (with sides parallel to the axes) in the frequency plane

$$\begin{eqnarray}(f,g)\mapsto \biggl(\mathop{\sum }_{\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}}\biggl|\int _{\mathbb{R}^{2}}\hat{f}(\unicode[STIX]{x1D709}){\hat{g}}(\unicode[STIX]{x1D702})\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})e^{2\unicode[STIX]{x1D70B}ix(\unicode[STIX]{x1D709}+\unicode[STIX]{x1D702})}\,d\unicode[STIX]{x1D709}\,d\unicode[STIX]{x1D702}\biggr|^{r}\biggr)^{1/r},\end{eqnarray}$$

$r>2$

$L^{p}\times L^{q}\rightarrow L^{s}$

$p,q,s^{\prime }$

$L^{r^{\prime }}$

$$\begin{eqnarray}\frac{1}{p}+\frac{1}{q}+\frac{1}{s^{\prime }}=1,\quad 0\leqslant \frac{1}{p},\frac{1}{q}<\frac{1}{r^{\prime }},\quad \text{and}\quad \frac{1}{s^{\prime }}<\frac{1}{r^{\prime }}.\end{eqnarray}$$

$s^{\prime }$

$L^{s}$

Following the ideas and methods of a recent work of Skinner and Urban, we prove the one divisibility of the Iwasawa main conjecture for nearly ordinary Hilbert modular forms under certain local hypotheses. As a consequence, we prove that for a Hilbert modular form of parallel weight, trivial character, and good ordinary reduction at all primes dividing $p$, if the central critical $L$-value is zero then the $p$-adic Selmer group of it has rank at least one. We also prove that one of the local assumptions in the main result of Skinner and Urban can be removed by a base-change trick.

This paper concerns involutions and anti-automorphisms on finite-dimensional associative algebras over a field. It is based on a survey talk given at the workshop on ‘Quadratic forms, algebras with involution and algebraic K-theory’ at the University of Nottingham, 13–15 May 2004. This workshop was part of the EU network HPRN-CT-2002-00287 on ‘Algebraic $K$-theory, linear algebraic groups and related structures’ and followed the London Mathematical Society Midlands Regional Meeting in Nottingham on 12 May, 2004.

The contents of the article are as follows:

1. Introduction and historical remarks;

2. Some general results;

3. Central simple algebras;

4. Group algebras.