For an irrational number $x\in [0,1)$, let $x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_{n}(x):n\geq 1\}$. Given $\unicode[STIX]{x1D6E9}\in \mathbb{N}$, for $n\geq 1$, the $n$th longest block function of $x$ with respect to $\unicode[STIX]{x1D6E9}$ is defined by $L_{n}(x,\unicode[STIX]{x1D6E9})=\max \{k\geq 1:a_{j+1}(x)=\cdots =a_{j+k}(x)=\unicode[STIX]{x1D6E9}~\text{for some}~j~\text{with}~0\leq j\leq n-k\}$, which represents the length of the longest consecutive sequence whose elements are all $\unicode[STIX]{x1D6E9}$ from the first $n$ partial quotients of $x$. We consider the growth rate of $L_{n}(x,\unicode[STIX]{x1D6E9})$ as $n\rightarrow \infty$ and calculate the Hausdorff dimensions of the level sets and exceptional sets arising from the longest block function.

We establish various new results on a problem proposed by Mahler [Some suggestions for further research. Bull. Aust. Math. Soc.29 (1984), 101–108] concerning rational approximation to fractal sets by rational numbers inside and outside the set in question. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich, and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover, we provide a non-trivial lower bound for the distance of a rational number $p/q$ outside the Cantor middle-third set $C$ to the set $C$, in terms of the denominator $q$. We further discuss patterns of rational numbers in fractal sets. We highlight two of them: firstly, an upper bound for the number of rational (algebraic) numbers in a fractal set up to a given height (and degree) for a wide class of fractal sets; and secondly, we find properties of the denominator structure of rational points in ‘missing-digit’ Cantor sets, generalizing claims of Nagy and Bloshchitsyn.

We present a complete characterisation of the radial asymptotics of degree-one Mahler functions as $z$ approaches roots of unity of degree $k^{n}$, where $k$ is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over $\mathbb{C}(z)$. Finally, we discuss asymptotic bounds towards generic points on the unit circle.

We give a new hypergeometric construction of rational approximations to ζ(4), which absorbs the earlier one from 2003 based on Bailey's 9F8 hypergeometric integrals. With the novel ingredients we are able to gain better control of the arithmetic and produce a record irrationality measure for ζ(4).

Let $F_{n}$ and $L_{n}$ be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in $s$ are defined by

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{F}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{F}^{\ast }(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{(-1)^{n+1}}{F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{L_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L}^{\ast }(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{(-1)^{n+1}}{L_{n}^{s}}}.\end{eqnarray}$$

$R(\unicode[STIX]{x1D70C}),Q(\unicode[STIX]{x1D70C}),$

$P(\unicode[STIX]{x1D70C})$

$\unicode[STIX]{x1D70C}$

$0<\unicode[STIX]{x1D70C}<1$

$s$

$m$

$$\begin{eqnarray}\mathop{\sum }_{s=1}^{m}(t_{s}\unicode[STIX]{x1D701}_{F}(2s)+u_{s}\unicode[STIX]{x1D701}_{F}^{\ast }(2s)+v_{s}\unicode[STIX]{x1D701}_{L}(2s)+w_{s}\unicode[STIX]{x1D701}_{L}^{\ast }(2s))=0\end{eqnarray}$$

$t_{s},u_{s},v_{s},w_{s}\in \mathbb{Q}$

$(1\leq s\leq m)$

$\mathbb{Q}$

$m$

$-2\unicode[STIX]{x1D701}_{F}(2)+\unicode[STIX]{x1D701}_{F}^{\ast }(2)+5\unicode[STIX]{x1D701}_{L}^{\ast }(2)=0$

$2s$

In this paper, we study the set

$$\begin{eqnarray}A=\{p(n)+2^{n}d~\text{mod}~1:n\geq 1\}\subset [0,1],\end{eqnarray}$$

$p$

$d$

$a\in [0,\infty )$

$a~\text{mod}~1$

$a$

$A$

Representations of the Cuntz algebra ${\mathcal{O}}_{N}$ are constructed from interval dynamical systems associated with slow continued fraction algorithms introduced by Giovanni Panti. Their irreducible decomposition formulas are characterized by using the modular group action on real numbers, as a generalization of results by Kawamura, Hayashi, and Lascu. Furthermore, a certain symmetry of such an interval dynamical system is interpreted as a covariant representation of the $C^{\ast }$-dynamical system of the “flip-flop” automorphism of ${\mathcal{O}}_{2}$.

We carry Sprindžuk’s classification of the complex numbers to the field $\mathbb{Q}_{p}$ of $p$-adic numbers. We establish several estimates for the $p$-adic distance between $p$-adic roots of integer polynomials, which we apply to show that almost all $p$-adic numbers, with respect to the Haar measure, are $p$-adic $\tilde{S}$-numbers of order 1.

In 1965, A. Livingston conjectured the $\overline{\mathbb{Q}}$-linear independence of logarithms of values of the sine function at rational arguments. In 2016, S. Pathak disproved the conjecture. In this article, we give a new proof of Livingston’s conjecture using some fundamental trigonometric identities. Moreover, we show that a stronger version of her theorem is true. In fact, we modify this conjecture by introducing a co-primality condition, and in that case we provide the necessary and sufficient conditions for the conjecture to be true. Finally, we identify a maximal linearly independent subset of the numbers considered in Livingston’s conjecture.

We give a partial answer to a question attributed to Chris Miller on algebraic values of certain transcendental functions of order less than one. We obtain $C(\log H)^{\unicode[STIX]{x1D702}}$ bounds for the number of algebraic points of height at most $H$ on certain subsets of the graphs of such functions. The constant $C$ and exponent $\unicode[STIX]{x1D702}$ depend on data associated with the functions and can be effectively computed from them.

Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.

We prove the analogue of the Ax–Lindemann–Weierstrass theorem for not necessarily arithmetic lattices of the automorphism group of the complex unit ball $\mathbb{B}^{n}$ using methods of several complex variables, algebraic geometry and Kähler geometry. Consider a torsion-free lattice $\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$ and the associated uniformization map $\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow \mathbb{B}^{n}/\unicode[STIX]{x1D6E4}=:X_{\unicode[STIX]{x1D6E4}}$. Given an algebraic subset $S\,\subset \,\mathbb{B}^{n}$ and writing $Z$ for the Zariski closure of $\unicode[STIX]{x1D70B}(S)$ in $X_{\unicode[STIX]{x1D6E4}}$ (which is equipped with a canonical quasi-projective structure), in some precise sense we realize $Z$ as a variety uniruled by images of algebraic subsets under the uniformization map, and study the asymptotic geometry of an irreducible component $\widetilde{Z}$ of $\unicode[STIX]{x1D70B}^{-1}(Z)$ as $\widetilde{Z}$ exits the boundary $\unicode[STIX]{x2202}\mathbb{B}^{n}$ by exploiting the strict pseudoconvexity of $\mathbb{B}^{n}$, culminating in the proof that $\widetilde{Z}\,\subset \,\mathbb{B}^{n}$ is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of $\text{ Aut}(\unicode[STIX]{x1D6FA})$ for (possibly reducible) bounded symmetric domains $\unicode[STIX]{x1D6FA}$.

In the field $\mathbb{K}$ of formal power series over a finite field $K$, we consider some lacunary power series with algebraic coefficients in a finite extension of $K(x)$. We show that the values of these series at nonzero algebraic arguments in $\mathbb{K}$ are $U$-numbers.

Let $\unicode[STIX]{x1D6F9}:[1,\infty )\rightarrow \mathbb{R}_{+}$ be a non-decreasing function, $a_{n}(x)$ the $n$th partial quotient of $x$ and $q_{n}(x)$ the denominator of the $n$th convergent. The set of $\unicode[STIX]{x1D6F9}$-Dirichlet non-improvable numbers,

$$\begin{eqnarray}G(\unicode[STIX]{x1D6F9}):=\{x\in [0,1):a_{n}(x)a_{n+1}(x)>\unicode[STIX]{x1D6F9}(q_{n}(x))\text{ for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$

$1/q^{2}\unicode[STIX]{x1D6F9}(q)$

${\mathcal{K}}(\unicode[STIX]{x1D6F9})$

${\mathcal{K}}(3\unicode[STIX]{x1D6F9})\subset G(\unicode[STIX]{x1D6F9})$

$s$

$s\in (0,1)$

$G(\unicode[STIX]{x1D6F9})\setminus {\mathcal{K}}(3\unicode[STIX]{x1D6F9})$

${\mathcal{K}}(\unicode[STIX]{x1D6F9})$

$G(\unicode[STIX]{x1D6F9})$

We give an integrability criterion on a real-valued non-increasing function $\unicode[STIX]{x1D713}$ guaranteeing that for almost all (or almost no) pairs $(A,\mathbf{b})$, where $A$ is a real $m\times n$ matrix and $\mathbf{b}\in \mathbb{R}^{m}$, the system

$$\begin{eqnarray}\Vert A\mathbf{q}+\mathbf{b}-\mathbf{p}\Vert ^{m}<\unicode[STIX]{x1D713}(T),\quad \Vert \mathbf{q}\Vert ^{n}<T,\end{eqnarray}$$

$\mathbf{p}\in \mathbb{Z}^{m}$

$\mathbf{q}\in \mathbb{Z}^{n}$

$T$

$\mathbb{R}^{m+n}$

$m=n=1$

We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are ultimately periodic.

Using methods from ergodic theory, we are able to partially resolve this conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial $p(n)$ with at least one irrational coefficient (except for the constant one) and integer $m\geqslant 2$, the sequence $\lfloor p(n)\rfloor \hspace{0.2em}{\rm mod}\hspace{0.2em}m$ is never automatic.

We also prove that the conjecture is equivalent to the claim that the set of powers of an integer $k\geqslant 2$ is not given by a generalised polynomial.

For $\unicode[STIX]{x1D6FC}$ an algebraic integer of any degree $n\geqslant 2$, it is known that the discriminants of the orders $\mathbb{Z}[\unicode[STIX]{x1D6FC}^{k}]$ go to infinity as $k$ goes to infinity. We give a short proof of this result.

Building upon ideas of the second and third authors, we prove that at least $2^{(1-\unicode[STIX]{x1D700})(\log s)/(\text{log}\log s)}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\unicode[STIX]{x1D700}$ is any positive real number and $s$ is large enough in terms of $\unicode[STIX]{x1D700}$. This lower bound is asymptotically larger than any power of $\log s$; it improves on the bound $(1-\unicode[STIX]{x1D700})(\log s)/(1+\log 2)$ that follows from the Ball–Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.

We prove the Hausdorff measure version of the matrix form of Gallagher’s theorem in the inhomogeneous setting, thereby proving a conjecture posed by Hussain and Simmons [‘The Hausdorff measure version of Gallagher’s theorem—closing the gap and beyond’, J. Number Theory186 (2018), 211–225].

In this paper, we provide two new extensions to a lemma of Bernik (1983). Applications are also discussed.