We continue the research of the relation $\hspace {1mm}\widetilde {\mid }\hspace {1mm}$ on the set $\beta \mathbb {N}$ of ultrafilters on $\mathbb {N}$, defined as an extension of the divisibility relation. It is a quasiorder, so we see it as an order on the set of $=_{\sim }$-equivalence classes, where $\mathcal {F}=_{\sim }\mathcal {G}$ means that $\mathcal {F}$ and $\mathcal {G}$ are mutually $\hspace {1mm}\widetilde {\mid }$-divisible. Here we introduce a new tool: a relation of congruence modulo an ultrafilter. We first recall the congruence of ultrafilters modulo an integer and show that $=_{\sim }$-equivalent ultrafilters do not necessarily have the same residue modulo $m\in \mathbb {N}$. Then we generalize this relation to congruence modulo an ultrafilter in a natural way. After that, using iterated nonstandard extensions, we introduce a stronger relation, which has nicer properties with respect to addition and multiplication of ultrafilters. Finally, we introduce a strengthening of $\hspace {1mm}\widetilde {\mid }\hspace {1mm}$ and show that it also behaves well with respect to the congruence relation.

Let p be a prime and let x be a p-adic integer. We prove two supercongruences for truncated series of the form

$$\begin{align*}\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots j_r^{}}\quad\mbox{and}\quad \sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{2}\cdots j_r^{2}}\end{align*}$$

For a polynomial $f(x)\in\mathbb{Q}[x]$ and rational numbers c, u, we put $f_c(x)\coloneqq f(x)+c$, and consider the Zsigmondy set $\calZ(f_c,u)$ associated to the sequence $\{f_c^n(u)-u\}_{n\geq 1}$, see Definition 1.1, where $f_c^n$ is the n-st iteration of fc. In this paper, we prove that if u is a rational critical point of f, then there exists an Mf > 0 such that $\mathbf M_f\geq \max_{c\in \mathbb{Q}}\{\#\calZ(f_c,u)\}$.

For any prime p, let $y(p)$ denote the smallest integer y such that every reduced residue class (mod p) is represented by the product of some subset of $\{1,\dots ,y\}$. It is easy to see that $y(p)$ is at least as large as the smallest quadratic nonresidue (mod p); we prove that $y(p) \ll _\varepsilon p^{1/(4 \sqrt e)+\varepsilon }$, thus strengthening Burgess’ classical result. This result is of intermediate strength between two other results, namely Burthe’s proof that the multiplicative group (mod p) is generated by the integers up to $O_\varepsilon (p^{1/(4 \sqrt e)+\varepsilon })$, and Munsch and Shparlinski’s result that every reduced residue class (mod p) is represented by the product of some subset of the primes up to $O_\varepsilon (p^{1/(4 \sqrt e)+\varepsilon })$. Unlike the latter result, our proof is elementary and similar in structure to Burgess’ proof for the least quadratic nonresidue.

The level of distribution of a complex-valued sequence $b$ measures the quality of distribution of $b$ along sparse arithmetic progressions $nd+a$. We prove that the Thue–Morse sequence has level of distribution $1$, which is essentially best possible. More precisely, this sequence gives one of the first nontrivial examples of a sequence satisfying a Bombieri–Vinogradov-type theorem for each exponent $\theta <1$. This result improves on the level of distribution $2/3$ obtained by Müllner and the author. As an application of our method, we show that the subsequence of the Thue–Morse sequence indexed by $\lfloor n^c\rfloor$, where $1 < c < 2$, is simply normal. This result improves on the range $1 < c < 3/2$ obtained by Müllner and the author and closes the gap that appeared when Mauduit and Rivat proved (in particular) that the Thue–Morse sequence along the squares is simply normal.

Given a positive integer M and $q \in (1, M+1]$ we consider expansions in base q for real numbers $x \in [0, {M}/{q-1}]$ over the alphabet $\{0, \ldots , M\}$. In particular, we study some dynamical properties of the natural occurring subshift $(\boldsymbol{{V}}_q, \sigma )$ related to unique expansions in such base q. We characterize the set of $q \in \mathcal {V} \subset (1,M+1]$ such that $(\boldsymbol{{V}}_q, \sigma )$ has the specification property and the set of $q \in \mathcal {V}$ such that $(\boldsymbol{{V}}_q, \sigma )$ is a synchronized subshift. Such properties are studied by analysing the combinatorial and dynamical properties of the quasi-greedy expansion of q. We also calculate the size of such classes as subsets of $\mathcal {V}$ giving similar results to those shown by Blanchard [10] and Schmeling in [36] in the context of $\beta $-transformations.

We construct eta-quotient representations of two families of q-series involving the Rogers–Ramanujan continued fraction by establishing related recurrence relations. We also display how these eta-quotient representations can be utilised to dissect certain q-series identities.

For positive integers n and d > 0, let $G(\mathbb {Q}^n,\; d)$ denote the graph whose vertices are the set of rational points $\mathbb {Q}^n$, with $u,v \in \mathbb {Q}^n$ being adjacent if and only if the Euclidean distance between u and v is equal to d. Such a graph is deemed “non-trivial” if d is actually realized as a distance between points of $\mathbb {Q}^n$. In this paper, we show that a space $\mathbb {Q}^n$ has the property that all pairs of non-trivial distance graphs $G(\mathbb {Q}^n,\; d_1)$ and $G(\mathbb {Q}^n,\; d_2)$ are isomorphic if and only if n is equal to 1, 2, or a multiple of 4. Along the way, we make a number of observations concerning the clique number of $G(\mathbb {Q}^n,\; d)$.

By making use of the ‘creative microscoping’ method, Guo and Zudilin [‘Dwork-type supercongruences through a creative $q$-microscope’, Preprint, 2020, arXiv:2001.02311] proved several Dwork-type supercongruences, including some conjectures of Swisher. In this paper, we apply the Guo–Zudilin method to prove a new Dwork-type supercongruence, which uniformly generalises several conjectures of Swisher.

For a positive real number $t$, define the harmonic continued fraction

$$\begin{eqnarray}\text{HCF}(t)=\biggl[\frac{t}{1},\frac{t}{2},\frac{t}{3},\ldots \biggr].\end{eqnarray}$$

$$\begin{eqnarray}\text{HCF}(t)=\frac{1}{1-2t(\frac{1}{t+2}-\frac{1}{t+4}+\frac{1}{t+6}-\cdots \,)}.\end{eqnarray}$$

Fix an alphabet $A=\{0,1,\ldots ,M\}$ with $M\in \mathbb{N}$. The univoque set $\mathscr{U}$ of bases $q\in (1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper describes how the points in the set $\mathscr{U}$ are distributed over the interval $(1,M+1)$ by determining the limit

$$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$

$q\in (1,M+1)$

$f(q)>0$

$q\in \overline{\mathscr{U}}\backslash \mathscr{C}$

$\mathscr{C}$

$f$

$\mathscr{U}$

$\mathscr{U}$

Erdös and Zaremba showed that $ \limsup_{n\to \infty} \frac{\Phi(n)}{(\log\log n)^2}=e^\gamma$, γ being Euler’s constant, where $\Phi(n)=\sum_{d|n} \frac{\log d}{d}$.

We extend this result to the function $\Psi(n)= \sum_{d|n} \frac{(\log d )(\log\log d)}{d}$ and some other functions. We show that $ \limsup_{n\to \infty}\, \frac{\Psi(n)}{(\log\log n)^2(\log\log\log n)}\,=\, e^\gamma$. The proof requires a new approach. As an application, we prove that for any $\eta>1$, any finite sequence of reals $\{c_k, k\in K\}$, $\sum_{k,\ell\in K} c_kc_\ell \, \frac{\gcd(k,\ell)^{2}}{k\ell} \le C(\eta) \sum_{\nu\in K} c_\nu^2(\log\log\log \nu)^\eta \Psi(\nu)$, where C(η) depends on η only. This improves a recent result obtained by the author.

We obtain a lower bound on the largest prime factor of the denominator of rational numbers in the Cantor set. This gives a stronger version of a recent result of Schleischitz [‘On intrinsic and extrinsic rational approximation to Cantor sets’, Ergodic Theory Dyn. Syst. to appear] obtained via a different argument.

We discuss a truncated identity of Euler and present a combinatorial proof of it. We also derive two finite identities as corollaries. As an application, we establish two related $q$-congruences for sums of $q$-Catalan numbers, one of which has been proved by Tauraso [‘$q$-Analogs of some congruences involving Catalan numbers’, Adv. Appl. Math. 48 (2012), 603–614] by a different method.

We introduce a notion of $q$-deformed rational numbers and $q$-deformed continued fractions. A $q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$-deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$-deformation of the Farey graph, matrix presentations and $q$-continuants are given, as well as a relation to the Jones polynomial of rational knots.

Under sufficiently strong assumptions about the first prime in an arithmetic progression, we prove that the number of Carmichael numbers up to $X$ is $\gg X^{1-R}$, where $R=(2+o(1))\log \log \log \log X/\text{log}\log \log X$. This is close to Pomerance’s conjectured density of $X^{1-R}$ with $R=(1+o(1))\log \log \log X/\text{log}\log X$.

In this note we use some $q$-congruences proved by the method of ‘creative microscoping’ to prove two conjectures on supercongruences involving central binomial coefficients. In particular, we confirm the $m=5$ case of Conjecture 1.1 of Guo [‘Some generalizations of a supercongruence of Van Hamme’, Integral Transforms Spec. Funct.28 (2017), 888–899].

We prove irregularities in the distribution of prime numbers in any Beatty sequence ${\mathcal{B}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ is a positive real irrational number of finite type.

A positive-definite diagonal quadratic form $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}\;(a_{1},\ldots ,a_{n}\in \mathbb{N})$ is said to be prime-universal if it is not universal and for every prime $p$ there are integers $x_{1},\ldots ,x_{n}$ such that $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}=p$. We determine all possible prime-universal ternary quadratic forms $ax^{2}+by^{2}+cz^{2}$ and all possible prime-universal quaternary quadratic forms $ax^{2}+by^{2}+cz^{2}+dw^{2}$. The prime-universal ternary forms are completely determined. The prime-universal quaternary forms are determined subject to the validity of two conjectures. We make no use of a result of Bhargava concerning quadratic forms representing primes which is stated but not proved in the literature.

For every integer $k\geq 2$ and every $A\subseteq \mathbb{N}$, we define the $k$-directions sets of $A$ as $D^{k}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{k}\}$ and $D^{\text{}\underline{k}}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{\text{}\underline{k}}\}$, where $\Vert \cdot \Vert$ is the Euclidean norm and $A^{\text{}\underline{k}}:=\{\boldsymbol{a}\in A^{k}:a_{i}\neq a_{j}\text{ for all }i\neq j\}$. Via an appropriate homeomorphism, $D^{k}(A)$ is a generalisation of the ratio set$R(A):=\{a/b:a,b\in A\}$. We study $D^{k}(A)$ and $D^{\text{}\underline{k}}(A)$ as subspaces of $S^{k-1}:=\{\boldsymbol{x}\in [0,1]^{k}:\Vert \boldsymbol{x}\Vert =1\}$. In particular, generalising a result of Bukor and Tóth, we provide a characterisation of the sets $X\subseteq S^{k-1}$ such that there exists $A\subseteq \mathbb{N}$ satisfying $D^{\text{}\underline{k}}(A)^{\prime }=X$, where $Y^{\prime }$ denotes the set of accumulation points of $Y$. Moreover, we provide a simple sufficient condition for $D^{k}(A)$ to be dense in $S^{k-1}$. We conclude with questions for further research.