Related papers: On Extending Pollard's Theorem for t-Representable…

Let $A = \{a_{1},a_{2},\dots{}\}$ $(a_{1} < a_{2} < \dots{})$ be an infinite sequence of nonnegative integers, and let $R_{A,2}(n)$ denote the number of solutions of $a_{x}+a_{y}=n$ $(a_{x},a_{y}\in A)$. P. Erd\H{o}s, A. S\'ark\"ozy and V.…

We show that if $\mathcal{L}_1$ and $\mathcal{L}_2$ are linear transformations from $\mathbb{Z}^d$ to $\mathbb{Z}^d$ satisfying certain mild conditions, then, for any finite subset $A$ of $\mathbb{Z}^d$, $$|\mathcal{L}_1 A+\mathcal{L}_2…

Let $G$ be a finite, non-trivial abelian group of exponent $m$, and suppose that $B_1, ..., B_k$ are generating subsets of $G$. We prove that if $k>2m \ln \log_2 |G|$, then the multiset union $B_1\cup...\cup B_k$ forms an additive basis of…

We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime residue field. The improvements are…

We prove that every set $A\subset\mathbb{Z}/p\mathbb{Z}$ with $\mathbb{E}_x\min(1_A*1_A(x),t)\le(2+\delta)t\mathbb{E}_x 1_A(a)$ is very close to an arithmetic progression. Here $p$ stands for a large prime and $\delta,t$ are small real…

Let $\mathcal{N}[k]$ be the multiset containing the $\binom{n-1}{k}$ products of $k$-subsets of $\{1,\ldots, n-1\}$. We show that if $n\geq (2c+3)^2$, then \begin{gather*}\left((-1)^c+\sum_{M\in \mathcal{N}[n-1-c]}M\right)\cdot(c+1)\equiv…

Let X = S \oplus G, where S is a countable abelian semigroup and G is a countably infinite abelian group such that {2g : g in G} is infinite. Let pi: X \to G be the projection map defined by pi(s,g) = g for all x =(s,g) in X. Let f:X \to…

We show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that qh has a definite determinantal representation. This is proved by considering sum-of-squares decompositions of certain bilinear forms…

Let $\alpha = (1+\sqrt{5})/2$ and define the lower and upper Wythoff sequences by $a_i = \lfloor i \alpha \rfloor$, $b_i = \lfloor i \alpha^2 \rfloor$ for $i \geq 1$. In a recent interesting paper, Kawsumarng et al. proved a number of…

We show a case of Zilber's Exponential-Algebraic Closedness Conjecture, establishing that the conjecture holds for varieties which split as the product of a linear subspace of the additive group $\mathbb{C}^n$ and an algebraic subvariety of…

In this paper, we study some supercongruences involving the sequence $$ t_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}2^k $$ and solve some open problems. For any odd prime $p$ and $p$-adic integer $x$, we determine…

For $p$ prime, $A \subseteq \mathbb{Z}/p\mathbb{Z}$ and $\lambda \in \mathbb{Z}$, the sum of dilates $A + \lambda \cdot A$ is defined by \[A + \lambda \cdot A = \{a + \lambda a' : a, a' \in A\}.\] The basic problem on such sums of dilates…

The $3k-4$ Theorem asserts that, if $A,\,B\subseteq \mathbb Z$ are finite, nonempty subsets with $|A|\geq |B|$ and $|A+B|=|A|+|B|+r< |A|+2|B|-3$, then there are arithmetic progressions $P_A$ and $P_B$ of common difference with $X\subseteq…

For two matrices $A$ and $B$, and large $n$, we show that most products of $n$ factors of $e^{A/n}$ and $n$ factors of $e^{B/n}$ are close to $e^{A + B}$. This extends the Lie-Trotter formula. The elementary proof is based on the relation…

We show that for any relatively prime integers $1\leq p<q$ and for any finite $A \subset \mathbb{Z}$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$$

Let $d\geq 2$ and $k\geq 1$ be fixed. We prove that, for every $\epsilon>0$ and every real $\beta$, there exist integers $1\leq b_1,\ldots,b_k\leq N$ such that \[ \left\|\sum_{j=1}^k b_j^{1/d}-\beta\right\| \ll_{d,k,\epsilon}…

Let A be a finite subset of a commutative additive group Z. The sumset and difference set of A are defined as the sets of pairwise sums and differences of elements of A, respectively. The well-known inequality $\sigma(A)^{1/2} \leq…

Let S be an abelian semigroup, written additively. Let A be a finite subset of S. We denote the cardinality of A by |A|. For any positive integer h, the sumset hA is the set of all sums of h not necessarily distinct elements of A. We define…

Let G be a branch group (as defined by Grigorchuk) acting on a tree T. A parabolic subgroup P is the stabiliser of an infinite geodesic ray in T. We denote by $\rho_{G/P}$ the associated quasi-regular representation. If G is discrete, these…

We show that for any prime prime $p\not=2$ $$\sum_{k=1}^{p-1} {(-1)^k\over k}{-{1\over 2} \choose k} \equiv -\sum_{k=1}^{(p-1)/2}{1\over k} \pmod{p^3}$$ by expressing the l.h.s. as a combination of alternating multiple harmonic sums.