Related papers: On the density of Kravitz sets

Let G=SL_3(Z/pZ), p a prime. Let A be a set of generators of G. Then A grows under the group operation. To be precise: denote by |S| the number of elements of a finite set S. Assume |A| < |G|^{1-\epsilon} for some \epsilon>0. Then |A\cdot…

Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every…

For set $A\subset {\mathbb {F}_p}^*$ define by ${\mathsf{sf}}(A)$ the size of the largest sum--free subset of $A.$ Alon and Kleitman showed that ${\mathsf{sf}} (A) \ge |A|/3+O(|A|/p).$ We prove that if ${\mathsf{sf}} (A)-|A|/3$ is small…

We investigate sumset decompositions of quite general sets with restricted prime factors. We manage to handle certain sets, such as the smooth numbers, even though they have little sieve amenability, and conclude that these sets cannot be…

In this article, we prove that for a completely multiplicative function $f$ from $\mathbb{N}^*$ to a field $K$ such that the set $$\{p \;|\; f(p)\neq 1_K \;\mbox{and }p \mbox{ is prime}\}$$ is finite, the asymptotic subword complexity of…

We prove the Arad-Herzog conjecture for various families of finite simple groups- if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are…

We establish a lower bound of 2/p(p-1) for the asymptotic density of the Motzkin numbers divisible by a general prime number p > 3. We provide a criteria for when this asymptotic density is actually 1. We also provide a partial…

We develop an elementary formula for certain non-trivial elements of upper cluster algebras. These elements have positive coefficients. We show that when the cluster algebra is acyclic these elements form a basis. Using this formula, we…

Every finite group whose order is divisible by a prime $p$ has at least $2 \sqrt{p-1}$ conjugacy classes.

In this paper we investigate some new problems in additive combinatorics. Our problems mainly involve permutations (or circular permutations) $n$ distinct numbers (or elements of an additive abelian group) $a_1,\ldots,a_n$ with adjacent…

We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if $t$ is an integer different from $0, 1$ or -1 and if $\A \subset \Zp$ is not too large (with respect to $p$),…

The classical Cauchy-Davenport theorem implies the lower bound n+1 for the number of distinct subsums that can be formed from a sequence of n elements of the cyclic group Z_p (when p is prime and n<p). We generalize this theorem to a…

Let $n$ be a positive integer and let $C_n$ be the cycle indicator of the symmetric group $S_n$. Carlitz proved that if $p$ is a prime, and if $r$ is a non negative integer, then we have the congruence $C_{r+np}\equiv (X_1^p-X_p)^nC_r…

Motivated by a problem in additive Ramsey theory, we extend Todorcevic's partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then…

Let $K$ be a cyclic totally real number field of odd degree over $\mathbb{Q}$ with odd class number, such that every totally positive unit is the square of a unit, and such that $2$ is inert in $K/\mathbb{Q}$. We define a family of number…

We prove that for an arbitrary $\kappa \le \frac{1}{3}$ any subset of $\mathbf{F}_p$ avoiding $t$ linear equations with three variables has size less than $O(p/t^\kappa)$. We also find several applications to problems about so--called…

For a cyclic group $a$, define the atom of $a$ as the set of all elements generating $a$. Given any two elements $a,b$ of a finite cyclic group $G$, we study the sumset of the atom of $a$ and the atom of $b$. It is known that such a sumset…

In this article, we shall generalize a theorem due to Frobenius in group theory, which asserts that if $p$ is a prime and $p^{r}$ divides the order of a finite group, then the number of subgroups of order $p^{r}$ is $\equiv$ 1(mod $p$).…

Denoting by Sigma(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Sigma(S)| >= |S|(|S|+2)/4-1 whenever S is symmetric, |G| is odd and Sigma(S) is aperiodic. Up to an additive constant of 2 this result is…

In 2003, H\'{e}thelyi and K\"{u}lshammer proposed that if $G$ is a finite group and $p$ is a prime dividing the group order, then $k(G)\geq 2\sqrt{p-1}$, and they proved this conjecture for solvable $G$ and showed that it is sharp for those…