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Some mean value theorems in the style of Bombieri-Vinogradov’s theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd $n \neg \equiv 1 (6)$ , Goldbach’s ternary problem $n = p_{1} + p_{2} + p_{3}$ is solvable with primes $p_{1}, p_{2}$ in short intervals $p_{i} \in [X_{i}, X_{i} + Y]$ with $X_{i}^{θ_{i}} = Y$ , $i = 1, 2$ , and $θ_{1}, θ_{2} \geq 0.933$ such that $(p_{1} + 2) (p_{2} + 2)$ has at most $9$ prime factors.

How Berger, Felzenbaum and Fraenkel revolutionized Covering Systems the same way that George Boole revolutionized Logic.

Zeilberger, Doron (2001)

The Electronic Journal of Combinatorics [electronic only]

How many points contain arithmetic progressions in their continued fraction expansion?

Xin Tong, Baowei Wang (2009)

Acta Arithmetica

Infinite families of noncototients

A. Flammenkamp, F. Luca (2000)

Colloquium Mathematicae

For any positive integer $n$ let ϕ(n) be the Euler function of n. A positive integer $n$ is called a noncototient if the equation x-ϕ(x)=n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression ${(2^{m} k)}_{m \geq 1}$ consists entirely of noncototients. We then use computations to detect seven such positive integers k.

Integers not of the form $c (2^{a} + 2^{b}) + p^{α}$

Pingzhi Yuan (2004)

Acta Arithmetica

La démonstration de Furstenberg du théorème de Szemerédi sur les progressions arithmétiques

Jean-Paul Thouvenot (1977/1978)

Séminaire Bourbaki

Large sets with small doubling modulo $p$ are well covered by an arithmetic progression

Oriol Serra, Gilles Zémor (2009)

Annales de l’institut Fourier

We prove that there is a small but fixed positive integer $ϵ$ such that for every prime $p$ larger than a fixed integer, every subset $S$ of the integers modulo $p$ which satisfies $| 2 S | \leq (2 + ϵ) | S |$ and $2 (| 2 S |) - 2 | S | + 3 \leq p$ is contained in an arithmetic progression of length $| 2 S | - | S | + 1$ . This is the first result of this nature which places no unnecessary restrictions on the size of $S$ .

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Discrepancy of cartesian products of arithmetic progressions.

Disjoint Beatty sequences.

Disjoint covering systems with precisely one multiple modulus

Double recurrence and almost sure convergence.

Důkaz věty, že existuje nekonečně mnoho kmenných čísel $(k p + 1)$ , je-li $p$ kmenné

Euler constants for arithmetic progressions

Exact m-covers and the linear form $\sum_{s = 1}^{k} x_{s} / n_{s}$

Fermat numbers and integers of the form $a^{k} + a^{l} + p^{α}$

Fifteen problems in number theory.

Fraenkel's partition and Brown's decomposition.

Goldbach’s problem with primes in arithmetic progressions and in short intervals