For a subset $A$ of an abelian group $G$, given its size $|A|$, its doubling $\kappa =|A+A|/|A|$, and a parameter $s$ which is small compared to $|A|$, we study the size of the largest sumset $A+A'$ that can be guaranteed for a subset $A'$ of $A$ of size at most $s$. We show that a subset $A'\subseteq A$ of size at most $s$ can be found so that $|A+A'| = \Omega (\!\min\! (\kappa ^{1/3},s)|A|)$. Thus, a sumset significantly larger than the Cauchy–Davenport bound can be guaranteed by a bounded size subset assuming that the doubling $\kappa$ is large. Building up on the same ideas, we resolve a conjecture of Bollobás, Leader and Tiba that for subsets $A,B$ of $\mathbb{F}_p$ of size at most $\alpha p$ for an appropriate constant $\alpha \gt 0$, one only needs three elements $b_1,b_2,b_3\in B$ to guarantee $|A+\{b_1,b_2,b_3\}|\ge |A|+|B|-1$. Allowing the use of larger subsets $A'$, we show that for sets $A$ of bounded doubling, one only needs a subset $A'$ with $o(|A|)$ elements to guarantee that $A+A'=A+A$. We also address another conjecture and a question raised by Bollobás, Leader and Tiba on high-dimensional analogues and sets whose sumset cannot be saturated by a bounded size subset.

Let p be a prime and let r, s be positive integers. In this paper, we prove that the Goormaghtigh equation $(x^m-1)/(x-1)=(y^n-1)/(y-1)$, $x,y,m,n \in {\mathbb {N}}$, $\min \{x,y\}>1$, $\min \{m,n\}>2$ with $(x,y)=(p^r,p^s+1)$ has only one solution $(x,y,m,n)=(2,5,5,3)$. This result is related to the existence of some partial difference sets in combinatorics.

We demonstrate that every difference set in a finite Abelian group is equivalent to a certain ‘regular’ covering of the lattice $ A_n = \{ \boldsymbol {x} \in \mathbb {Z} ^{n+1} : \sum _{i} x_i = 0 \} $ with balls of radius $ 2 $ under the $ \ell _1 $ metric (or, equivalently, a covering of the integer lattice $ \mathbb {Z} ^n $ with balls of radius $ 1 $ under a slightly different metric). For planar difference sets, the covering is also a packing, and therefore a tiling, of $ A_n $ . This observation leads to a geometric reformulation of the prime power conjecture and of other statements involving Abelian difference sets.

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any finite set of numbers $B.$ The bound is tight up to the constant multiplier. We give a new proof to this result using bounds on crossing numbers of geometric graphs. We construct examples showing the limits of possible improvements. In particular, we show that there are arbitrarily large sets with different consecutive differences and sub-quadratic sumset size.

Let $A$ be a set of natural numbers. Recent work has suggested a strong link between the additive energy of $A$ (the number of solutions to $a_{1}+a_{2}=a_{3}+a_{4}$ with $a_{i}\in A$) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of $A$ modulo $1$. There appears to be reasonable evidence to speculate a sharp Khinchin-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian.

We show that if a Barker sequence of length $n>13$ exists, then either n $=$ 3 979 201 339 721 749 133 016 171 583 224 100, or $n > 4\cdot 10^{33}$. This improves the lower bound on the length of a long Barker sequence by a factor of nearly $2000$. We also obtain eighteen additional integers $n<10^{50}$ that cannot be ruled out as the length of a Barker sequence, and find more than 237 000 additional candidates $n<10^{100}$. These results are obtained by completing extensive searches for Wieferich prime pairs and using them, together with a number of arithmetic restrictions on $n$, to construct qualifying integers below a given bound. We also report on some updated computations regarding open cases of the circulant Hadamard matrix problem.

We consider the problem of sums of dilates in groups of prime order. It is well known that sets with small density and small sumset in p behave like integer sets. Thus, given A ⊂ p of sufficiently small density, it is straightforward to show that

\begin{linenomath} $$| \lambda_{1}A+\lambda_{2}A+\cdots+ \lambda_{k}A | \ge\biggl(\sum_{i}|\lambda_{i}|\biggr)|A|- o(|A|).$$ \end{linenomath}

Let $v\,\ge \,k\,\ge \,1$ and $\lambda \,\ge \,0$ be integers. A block design$\text{BD}\left( v,\,k,\,\lambda \right)$ is a collection $\mathcal{A}$ of $k$-subsets of a $v$-set $X$ in which every unordered pair of elements from $X$ is contained in exactly $\lambda $ elements of $\mathcal{A}$. More generally, for a fixed simple graph $G$, a graph design$\text{GD}\left( v,\,G,\,\lambda \right)$ is a collection $\mathcal{A}$ of graphs isomorphic to $G$ with vertices in $X$ such that every unordered pair of elements from $X$ is an edge of exactly $\lambda $ elements of $\mathcal{A}$. A famous result of Wilson says that for a fixed $ $ and $\lambda $, there exists a $\text{GD}\left( v,\,G,\,\lambda \right)$ for all sufficiently large $ $ satisfying certain necessary conditions. A block (graph) design as above is resolvable if $\mathcal{A}$ can be partitioned into partitions of (graphs whose vertex sets partition) $X$. Lu has shown asymptotic existence in $v$ of resolvable $\text{BD}\left( v,\,k,\,\lambda \right)$, yet for over twenty years the analogous problem for resolvable $\text{GD}\left( v,\,G,\,\lambda \right)$ has remained open. In this paper, we settle asymptotic existence of resolvable graph designs.

Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, the first known family of partial difference sets with negative Latin square type parameters is constructed in nonelementary abelian groups, the groups $\Z_4^{2k}\times \Z_2^{4 \ell-4k}$ for all $k$ when $\ell$ is odd and for all $k < \ell$ when $\ell$ is even. Similarly, partial difference sets with Latin square type parameters are constructed in the same groups for all $k$ when $\ell$ is even and for all $k<\ell$ when $\ell$ is odd. These constructions provide the first example where the non-homomorphic bijection approach outlined by Hagita and Schmidt can produce difference sets in groups that previously had no known constructions. Computer computations indicate that the strongly regular graphs associated to the partial difference sets are not isomorphic to the known graphs, and it is conjectured that the family of strongly regular graphs will be new.