Related papers: Difference sets and power residues

Let $\mathbb{F}_p$ be the field of a prime order $p.$ It is known that for any integer $N\in [1,p]$ one can construct a subset $A\subset\mathbb{F}_p$ with $|A|= N$ such that $$ \max\{|A+A|, |AA|\}\ll p^{1/2}|A|^{1/2}. $$ In the present…

A composite positive integer $n$ has the Lehmer property if $\phi(n)$ divides $n-1,$ where $\phi$ is an Euler totient function. In this note we shall prove that if $n$ has the Lehmer property, then $n\leq 2^{2^{K}}-2^{2^{K-1}}$, where $K$…

It is well known that $\sum_{p\le n} 1/p =\ln(\ln(n)) + O(1)$ where $p$ goes over the primes. We give several known proofs of this. We first present a a proof that $\ge \ln(\ln(n)) + O(1)$. This is based on Euler's proof that $\sum_p 1/p$…

We show that if $A=\{a_1 < a_2 < \ldots < a_k\}$ is a set of real numbers such that the differences of the consecutive elements are distinct, then for and finite $B \subset \mathbb{R}$, $$|A+B|\gg |A|^{1/2}|B|.$$ The bound is tight up to…

Let $p_n$ denote the $n$-th prime number, $\{q_n\}$ be a sequence of positive numbers and $x\in\mathbb{R}$. In this note we prove that the inequality $$q_n p_{n+1}^{x}-q_{n+1}p_{n}^{x}<p_{n}^{x}p_{n+1}^{x-1}, $$ holds for infinitely many…

Consider the discrete maximal function acting on finitely supported functions on the integers, \[ \mathcal{C}_\Lambda f(n) := \sup_{\lambda \in \Lambda} | \sum_{p \in \pm \mathbb{P}} f(n-p) \log |p| \frac{e^{2\pi i \lambda p}}{p} |,\] where…

We show that if A is a set having small subtractive doubling in an abelian group, that is |A-A|< K|A|, then there is a polynomially large subset B of A-A so that the additive energy of B is large than (1/K)^{1 - \epsilon) where epsilon is a…

Let A and B be finite sets in a commutative group. We bound |A+hB| in terms of |A|, |A+B| and h. We provide a submultiplicative upper bound that improves on the existing bound of Imre Ruzsa by inserting a factor that decreases with h.

Nagell proved that for each prime $p\equiv 1\pmod{3}$, $p > 7$, there is a prime $q<2p^{1/2}$ that is a cubic residue modulo $p$. Here we show that for each fixed $\epsilon > 0$, and each prime $p\equiv 1\pmod{3}$ with $p > p_0(\epsilon)$,…

Given a prime number $l$ and a finite set of integers $S=\{a_1,...,a_m\}$ we find out the exact degree of the extension $\mathbb{Q}(a_1^{\frac{1}{l}},...,a_m^{\frac{1}{l}})/\mathbb{Q}$. We give an algorithm to compute this degree and then…

We find an upper bound for the sum $\sum_{x<n\leq 2x}\textbf{1}_{\mathbb{P}}(n+h_{i_{1}})\cdots\textbf{1}_{\mathbb{P}}(n+h_{i_{m+1}})w_{n}$, where $(h_{i_{1}},...,h_{i_{m+1}})$ is any $(m+1)$-tuple of elements in the admissible set…

We look at upper bounds for the count of certain primes related to the Fermat numbers $F_n=2^{2^n}+1$ called elite primes. We first note an oversight in a result of Krizek, Luca and Somer and give the corrected, slightly weaker upper bound.…

For a finite set $A\subset \mathbb{R}$ and real $\lambda$, let $A+\lambda A:=\{a+\lambda b :\, a,b\in A\}$. Combining a structural theorem of Freiman on sets with small doubling constants together with a discrete analogue of…

For any measure preserving system $(X,\mathcal{X},\mu,T)$ and $A\in\mathcal{X}$ with $\mu(A)>0$, we show that there exist infinitely many primes $p$ such that $\mu\bigl(A\cap T^{-(p-1)}A\cap T^{-2(p-1)}A\bigr) > 0$ (the same holds with…

We give exponential upper bounds for $P(S \le k)$, in particular $P(S=0)$, where $S$ is a sum of indicator random variables that are positively associated. These bounds allow, in particular, a comparison with the independent case. We give…

We study subsets of $\mathbb{F}_p^n$ that do not contain progressions of length $k$. We denote by $r_k(\mathbb{F}_p^n)$ the cardinality of such subsets containing a maximal number of elements. In this paper we focus on the case $k=p$ and…

We show that any subset of $\mathbb{Z}_p^n$ ($p$ an odd prime) without $3$-term arithmetic progression has size $O(p^{cn})$, where $c:=1-\frac{1}{18\log p}<1$. In particular, we find an upper bound of $O(2.84^n)$ on the maximum size of an…

For a prime $p$ and a matrix $A \in \mathbb{Z}^{n \times n}$, write $A$ as $A = p (A \,\mathrm{quo}\, p) + (A \,\mathrm{rem}\, p)$ where the remainder and quotient operations are applied element-wise. Write the $p$-adic expansion of $A$ as…

We give upper bounds for triples of subsets of a finite group such that the triples of elements that multiply to 1 form a perfect matching. Our bounds are the first to give exponential savings in powers of an arbitrary finite group.…

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in…