A set A of positive integers is a Bh-set if all the sums of the form a1 + . . . + ah, with a1,. . .,ah ∈ A and a1 ⩽ . . . ⩽ ah, are distinct. We provide asymptotic bounds for the number of Bh-sets of a given cardinality contained in the interval [n] = {1,. . .,n}. As a consequence of our results, we address a problem of Cameron and Erdős (1990) in the context of Bh-sets. We also use these results to estimate the maximum size of a Bh-sets contained in a typical (random) subset of [n] with a given cardinality.

We show that for any coprime integers λ1, . . ., λk and any finite A ⊂ $\mathbb{Z}$ , one has

|\lambda_1 \cdot A + \cdots + \lambda_k \cdot A| \geq (|\lambda_1| + \cdots + |\lambda_k|)|A|- C,

We demonstrate that many properties of topological spaces connected with the notion of resolvability are preserved by the relation of similarity between topologies. Moreover, many of them can be characterised by the properties of the algebra of sets with nowhere dense boundary and the ideal of nowhere dense sets. We use these results to investigate whether a given pair of an algebra and an ideal is topological.

Let $g\geq 2$ be a fixed integer. Let $\mathbb{N}$ denote the set of all nonnegative integers and let $A$ be a subset of $\mathbb{N}$. Write $r_{2}(A,n)=\sharp \{(a_{1},a_{2})\in A^{2}:a_{1}+a_{2}=n\}.$ We construct a thin, strongly minimal, asymptotic $g$-adic basis $A$ of order two such that the set of $n$ with $r_{2}(A,n)=2$ has density one.

Let $\mathbb{A}=(A,+)$ be a (possibly non-commutative) semigroup. For $Z\subseteq A$, we define $Z^{\times }:=Z\cap \mathbb{A}^{\times }$, where $\mathbb{A}^{\times }$ is the set of the units of $\mathbb{A}$ and

$$\begin{eqnarray}{\it\gamma}(Z):=\sup _{z_{0}\in Z^{\times }}\inf _{z_{0}\neq z\in Z}\text{\text{ord}}(z-z_{0}).\end{eqnarray}$$

${\it\gamma}(\cdot )$

$\mathbb{A}$

$X,Y\subseteq A$

$$\begin{eqnarray}|X+Y|\geqslant \min ({\it\gamma}(X+Y),|X|+|Y|-1).\end{eqnarray}$$

$\mathbb{A}$

${\it\gamma}(X+Y)$

$|S|$

$S$

$\mathbb{A}$

Let $G$ be a finite abelian group and $A\subseteq G$. For $n\in G$, denote by $r_{A}(n)$ the number of ordered pairs $(a_{1},a_{2})\in A^{2}$ such that $a_{1}+a_{2}=n$. Among other things, we prove that for any odd number $t\geq 3$, it is not possible to partition $G$ into $t$ disjoint sets $A_{1},A_{2},\dots ,A_{t}$ with $r_{A_{1}}=r_{A_{2}}=\cdots =r_{A_{t}}$.

We study analytic properties of certain infinite products of cyclotomic polynomials that generalise some products introduced by Mahler. We characterise those that have the unit circle as a natural boundary and use associated Dirichlet series to obtain their asymptotic behaviour near roots of unity.

The Frobenius number $F(\boldsymbol{a})$ of a lattice point $\boldsymbol{a}$ in $\mathbb{R}^{d}$ with positive coprime coordinates, is the largest integer which can not be expressed as a non-negative integer linear combination of the coordinates of $\boldsymbol{a}$. Marklof in [The asymptotic distribution of Frobenius numbers, Invent. Math. 181 (2010), 179–207] proved the existence of the limit distribution of the Frobenius numbers, when $\boldsymbol{a}$ is taken to be random in an enlarging domain in $\mathbb{R}^{d}$. We will show that if the domain has piecewise smooth boundary, the error term for the convergence of the distribution function is at most a polynomial in the enlarging factor.

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf{c}(H)$ of $H$ is the smallest integer $N$ with the following property: for each $a\in H$ and each pair of factorizations $z,z^{\prime }$ of $a$ , there exist factorizations $z=z_{0},\dots ,z_{k}=z^{\prime }$ of $a$ such that, for each $i\in [1,k]$ , $z_{i}$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. To exclude trivial cases, suppose that $|G|\geq 3$ . Then the catenary degree depends only on the class group $G$ and we have $\mathsf{c}(H)\in [3,\mathsf{D}(G)]$ , where $\mathsf{D}(G)$ denotes the Davenport constant of $G$ . The cases when $\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$ have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldinger et al. [‘The catenary degree of Krull monoids I’, J. Théor. Nombres Bordeaux23 (2011), 137–169], we determine the class groups satisfying $\mathsf{c}(H)=\mathsf{D}(G)-1$ . Apart from the extremal cases mentioned, the precise value of $\mathsf{c}(H)$ is known for no further class groups.

This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations $af(z^{r})=f(z)+R(z)$ , where $a$ is a nonzero complex number, $r$ an integer greater than 1, and $R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained.

Let $A\subset \{1,\dots ,N\}$ be a set of prime numbers containing no non-trivial arithmetic progressions. Suppose that $A$ has relative density ${\it\alpha}=|A|/{\it\pi}(N)$, where ${\it\pi}(N)$ denotes the number of primes in the set $\{1,\dots ,N\}$. By modifying Helfgott and De Roton’s work [Improving Roth’s theorem in the primes. Int. Math. Res. Not. IMRN2011 (4) (2011), 767–783], we improve their bound and show that

$$\begin{eqnarray}{\it\alpha}\ll \frac{(\log \log \log N)^{6}}{\log \log N}.\end{eqnarray}$$

We obtain density estimates for subsets of the $n$-dimensional integer lattice lacking four-term matrix progressions. As a consequence, we show that a subset of the grid $\{1,2,\dots ,N\}^{2}$ lacking four corners in a square has size at most $\mathit{CN}^{2}(\log \log N)^{-c}$. Our proofs involve the density increment method of Roth [J. London Math. Soc.28 (1953), 104–109] and Gowers [Geom. Funct. Anal.11(3) (2001), 465–588], together with the $U^{3}$-inverse theorem of Green and Tao [Proc. Edinb. Math. Soc. (2) 51(1) (2008), 73–153].

In this paper we present a new family of identities for multiple harmonic sums which generalize a recent result of Hessami Pilehrood et al  [Trans. Amer. Math. Soc. (to appear)]. We then apply it to obtain a family of identities relating multiple zeta star values to alternating Euler sums. In such a typical identity the entries of the multiple zeta star values consist of blocks of arbitrarily long 2-strings separated by positive integers greater than two while the largest depth of the alternating Euler sums depends only on the number of 2-string blocks but not on their lengths.

We determine a lower gap property for the growth of an unbounded $\mathbb{Z}$-valued $k$-regular sequence. In particular, if $f:\mathbb{N}\to \mathbb{Z}$ is an unbounded $k$-regular sequence, we show that there is a constant $c>0$ such that $|f(n)|>c\log n$ infinitely often. We end our paper by answering a question of Borwein, Choi and Coons on the sums of completely multiplicative automatic functions.

The cubic version of the Lucas cryptosystem is set up based on the cubic recurrence relation of the Lucas function by Said and Loxton [‘A cubic analogue of the RSA cryptosystem’, Bull. Aust. Math. Soc.68 (2003), 21–38]. To implement this type of cryptosystem in a limited environment, it is necessary to accelerate encryption and decryption procedures. Therefore, this paper concentrates on improving the computation time of encryption and decryption in cubic Lucas cryptosystems. The new algorithm is designed based on new properties of the cubic Lucas function and mathematical techniques. To illustrate the efficiency of our algorithm, an analysis was carried out with different size parameters and the performance of the proposed and previously existing algorithms was evaluated with experimental data and mathematical analysis.

Let $Q(N;q,a)$ be the number of squares in the arithmetic progression $qn+a$ , for $n=0$ , $1,\ldots,N-1$ , and let $Q(N)$ be the maximum of $Q(N;q,a)$ over all non-trivial arithmetic progressions $qn + a$ . Rudin’s conjecture claims that $Q(N)=O(\sqrt{N})$ , and in its stronger form that $Q(N)=Q(N;24,1)$ if $N\ge 6$ . We prove the conjecture above for $6\le N\le 52$ . We even prove that the arithmetic progression $24n+1$ is the only one, up to equivalence, that contains $Q(N)$ squares for the values of $N$ such that $Q(N)$ increases, for $7\le N\le 52$ ( $N=8,13,16,23,27,36,41$ and $52$ ).

Supplementary materials are available with this article.

Let $P$ and $Q$ be non-zero integers. The generalized Fibonacci sequence $\{ {U}_{n} \} $ and Lucas sequence $\{ {V}_{n} \} $ are defined by ${U}_{0} = 0$, ${U}_{1} = 1$ and ${U}_{n+ 1} = P{U}_{n} + Q{U}_{n- 1} $ for $n\geq 1$ and ${V}_{0} = 2, {V}_{1} = P$ and ${V}_{n+ 1} = P{V}_{n} + Q{V}_{n- 1} $ for $n\geq 1$, respectively. In this paper, we assume that $Q= 1$. Firstly, we determine indices $n$ such that ${V}_{n} = k{x}^{2} $ when $k\vert P$ and $P$ is odd. Then, when $P$ is odd, we show that there are no solutions of the equation ${V}_{n} = 3\square $ for $n\gt 2$. Moreover, we show that the equation ${V}_{n} = 6\square $ has no solution when $P$ is odd. Lastly, we consider the equations ${V}_{n} = 3{V}_{m} \square $ and ${V}_{n} = 6{V}_{m} \square $. It has been shown that the equation ${V}_{n} = 3{V}_{m} \square $ has a solution when $n= 3, m= 1$, and $P$ is odd. It has also been shown that the equation ${V}_{n} = 6{V}_{m} \square $ has a solution only when $n= 6$. We also solve the equations ${V}_{n} = 3\square $ and ${V}_{n} = 3{V}_{m} \square $ under some assumptions when $P$ is even.

We prove that ${ \mathbb{Z} }_{{p}^{n} } $ and ${ \mathbb{Z} }_{p} [t] / ({t}^{n} )$ are polynomially equivalent if and only if $n\leq 2$ or ${p}^{n} = 8$. For the proof, employing Bernoulli numbers, we explicitly provide the polynomials which compute the carry-on part for the addition and multiplication in base $p$. As a corollary, we characterize finite rings of ${p}^{2} $ elements up to polynomial equivalence.

For $n\in \mathbb{Z} $ and $A\subseteq \mathbb{Z} $, let ${r}_{A} (n)= \# \{ ({a}_{1} , {a}_{2} )\in {A}^{2} : n= {a}_{1} + {a}_{2} , {a}_{1} \leq {a}_{2} \} $ and ${\delta }_{A} (n)= \# \{ ({a}_{1} , {a}_{2} )\in {A}^{2} : n= {a}_{1} - {a}_{2} \} $. We call $A$ a unique representation bi-basis if ${r}_{A} (n)= 1$ for all $n\in \mathbb{Z} $ and ${\delta }_{A} (n)= 1$ for all $n\in \mathbb{Z} \setminus \{ 0\} $. In this paper, we construct a unique representation bi-basis of $ \mathbb{Z} $ whose growth is logarithmic.

We study the mixing properties of progressions $(x, xg, x{g}^{2} )$ , $(x, xg, x{g}^{2} , x{g}^{3} )$ of length three and four in a model class of finite nonabelian groups, namely the special linear groups ${\mathrm{SL} }_{d} (F)$ over a finite field $F$ , with $d$ bounded. For length three progressions $(x, xg, x{g}^{2} )$ , we establish a strong mixing property (with an error term that decays polynomially in the order $\vert F\vert $ of $F$ ), which among other things counts the number of such progressions in any given dense subset $A$ of ${\mathrm{SL} }_{d} (F)$ , answering a question of Gowers for this class of groups. For length four progressions $(x, xg, x{g}^{2} , x{g}^{3} )$ , we establish a partial result in the $d= 2$ case if the shift $g$ is restricted to be diagonalizable over $F$ , although in this case we do not recover polynomial bounds in the error term. Our methods include the use of the Cauchy–Schwarz inequality, the abelian Fourier transform, the Lang–Weil bound for the number of points in an algebraic variety over a finite field, some algebraic geometry, and (in the case of length four progressions) the multidimensional Szemerédi theorem.