Related papers: Rearranging small sets for distinct partial sums

Let $p$ be an odd prime and $A \subseteq \mathbb{F}_p$ be a subset of the finite field with $p$ elements. We show that $A \times A \subseteq \mathbb{F}_p^2$ determines at least a constant multiple of $\min\{p, |A|^{3/2}\}$ distinct pinned…

Let $f(d)$ be the smallest number so that every set in $R^d$ of diameter 1 can be partitioned into $f(d)$ sets of diameter smaller than 1. Borsuk's conjecture was that $f(d)\! =\!d\!+\!1$. We prove that $f(d)\! \ge\! (1.2)^{\sqrt d}$ for…

Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain constructions of expanders, we characterize all polynomials $P$…

Let $\mathcal{P}$ denote the set of all primes. In 1950, P. Erd\H{o}s conjectured that if $c$ is an arbitrarily given constant, $x$ is sufficiently large and $a_1,\dots , a_t$ are positive integers with $a_1<a_2<\cdot\cdot\cdot<a_t\leqslant…

Let G be any group and $a_1G_1,...,a_kG_k (k>1)$ be left cosets in G. In 1974 Herzog and Sch\"onheim conjectured that if $\Cal A=\{a_iG_i\}_{i=1}^k$ is a partition of G then the (finite) indices $n_1=[G:G_1],...,n_k=[G:G_k]$ cannot be…

A complete theory ${\mathcal T}$ of partial order is an FLD$_1$-theory iff some (equivalently, any) of its models ${\mathbb X}$ admits a finite lexicographic decomposition ${\mathbb X} =\sum _{{\mathbb I}}{\mathbb X} _i$, where ${\mathbb…

In analytic number theory, several results make use of information regarding the prime values of a multiplicative function in order to extract information about its averages. Examples of such results include Wirsing's theorem and the…

We define two notions of partial sums of a Riordan array, corresponding respectively to the partial sums of the rows and the partial sums of the columns of the Riordan array in question. We characterize the matrices that arise from these…

Suppose that G is an abelian group and A is a finite subset of G containing no three-term arithmetic progressions. We show that |A+A| >> |A|(log |A|)^{1/3-\epsilon} for all \epsilon>0.

A well-known conjecture in analytic number theory states that for every pair of sets $X,Y\subset\mathbb{Z}/p\mathbb{Z}$, each of size at least $\log ^C p$ (for some constant $C$) we have that the number of pairs $(x,y)\in X\times Y$ such…

The Herzog-Sch\"onheim conjecture states that if $H_1, \ldots, H_k$ are subgroups of a group $G$ and $x_1, \ldots, x_k$ are elements of $G$ such that $H_1x_1, \ldots, H_kx_k$ is a partition of $G$ into cosets, then two of these subgroups…

We generalize the geometric sequence $\{a^p, a^{p-1}b, a^{p-2}b^2,...,b^p\}$ to allow the $p$ copies of $a$ (resp. $b$) to all be different. We call the sequence $\{a_1a_2a_3\cdots a_p, b_1a_2a_3\cdots a_p, b_1b_2a_3\cdots a_p,\ldots,…

Let $S$ be a Scott set, or even an $\omega$-model of $\mathsf{WWKL}$. Then for each $A\in S$, either there is $X \in S$ that is weakly 2-random relative to $A$, or there is $X\in S$ that is 1-generic relative to $A$. It follows that if…

We help Alice play a certain "convergence game" against Bob and win the prize, which is a constructive solution to a problem by Erd\H{o}s and Graham, posed in their 1980 book on open questions in combinatorial number theory. Namely, after…

This is an English translation of Euler's ``Theoremata circa residua ex divisione potestatum relicta'', Novi Commentarii academiae scientiarum Petropolitanae 7 (1761), 49-82. E262 in the Enestrom index. Euler gives many elementary results…

We study the extent to which sets A in Z/NZ, N prime, resemble sets of integers from the additive point of view (``up to Freiman isomorphism''). We give a direct proof of a result of Freiman, namely that if |A + A| < K|A| and |A| < c(K)N…

This work is about a partition problem which is an instance of the distance magic graph labeling problem. Given positive integers $n,k$ and $p_1\le p_2\le \cdots\le p_k$ such that $p_1+\cdots+p_k=n$ and $k$ divides $\sum_{i=1}^ni$, we study…

We discuss several questions concerning sum-free sets in groups, raised by Erd\H{o}s in his survey "Extremal problems in number theory" (Proceedings of the Symp. Pure Math. VIII AMS) published in 1965. Among other things, we give a…

Many combinatorial sequences (for example, the Catalan and Motzkin numbers) may be expressed as the constant term of $P(x)^k Q(x)$, for some Laurent polynomials $P(x)$ and $Q(x)$ in the variable $x$ with integer coefficients. Denoting such…

We use topological ideas to show that, assuming the conjecture of Erd\"(o)s on subsets of positive integers having no $p$ terms in arithmetic progression (A. P.), there must exist a subset $M_p$ of positive integers with no $p$ terms in A.…