**10**

votes

**0**answers

575 views

### What are the limits of the Erdős-Rankin method for covering intervals by arithmetic progressions?

To construct gaps between primes which are marginally larger than average, Erdős and Rankin covered an interval $[1,y]$ with arithmetic progressions with prime differences. A nice short exposition is ...

**6**

votes

**0**answers

337 views

### Large sets not containing arithmetic progressions of length 3 in intervals

Given a large enough natural number $N$, let $\Delta_N=\{A \subseteq [N, 2N]: A$ contains no arithmetic progressions of length $3 \},$ where for natural numbers $N<M$ we have $[N, M]=\{N, N+1, ..., ...

**4**

votes

**0**answers

226 views

### The original proof of Szemerédi's Theorem

Nowadays there are plenty of different proofs of the celebrated Szemerédi's Theorem but for historical reasons I would like to read and understand the original proof. The proof is very tricky and, for ...

**2**

votes

**0**answers

967 views

### Fun question in additive combinatorics

It is easy to see that for a finite set of integers $A$ of cardinality $n$, the cardinality of the sumset $A+A$ satisfies
$$
2n-1\leq |A+A|\leq \frac{n(n+1)}{2}.
$$
The lower bound is essentially ...

**1**

vote

**0**answers

80 views

### Consecutive integers divisible by consecutive small numbers

Given $n$, what is the largest set of consecutive integers in $[n,2n]$ can we have so that each integer is divisible by a distinct element from $[\log n,2\log n]$ (no partiular order)? So apriori I am ...

**0**

votes

**0**answers

43 views

### Arithmetic progressions on a graph

Given $K_k$ a complete graph what is the minimum $n\in\Bbb N$ needed so that there is a map:
$$f:\{0,1,\dots,n\}\rightarrow\mathsf{Edges}(K_k)$$ which makes every simple cycle to be $r$-term ...

**0**

votes

**0**answers

45 views

### What is the maximum possible size of a $k$-average free subset of $[1, \dots, n]$?

Say that a subset $S \subset [1, \dots, n]$ is $k$-average free if, for all multi-subsets $A$ of $S$ with $|A| \le k$, the arithmetic mean of $A$ is not in $A$ (except for the trivial case when all ...

**0**

votes

**0**answers

152 views

### arithmetic progressions with few primes

Is this true ?
Let $\beta_0$ be a positive number. One may find $\beta>\beta_0$, $0<\lambda<1$, and infinitely many $q>1$ so that there exists an arithmetic progression of step $q$, $a_1, ...

**0**

votes

**0**answers

315 views

### Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by:
$$
\ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1.
$$
For all terms of $A$ greater than $\ \...

**0**

votes

**0**answers

116 views

### Arithmetic progression and 3^m,3^{m+1} intervals

I'm trying to prove (or disprove) the following "conjecture".Given the following set of powers of two:
$$A = \{ x \mid x = 2^n \text{ and } 2^{n-1} < 3^m < x < 3^{m+1} < 2^{n+1}\}$$
(...

**0**

votes

**0**answers

211 views

### Conjecture about distribution of primes in arithmetic progression

For my work, i need the following
Conjecture: Let $N$ large number such that exist a prime number $q$ and $A>\frac{1}{2}$ such that $N^{1/2}<N^{A}\leq q-1<N.$ Then $\forall a\in\left[1,\, q\...

**0**

votes

**0**answers

294 views

### Greedy sequences without k-term arithmetic progressions

If $S_k$ is the greedy sequence with no length-k arithmetic subsequence, (ie $S_3$ = A003278 , $S_4$ = A005837 , $S_5$ = A020655 ), is it guaranteed that any other sequence $a$ with no length-k ...

**-1**

votes

**0**answers

92 views

### A question about arithmetic progressions and prime numbers

"I took number $3$ and observed:
$3$ is an arithmetic progression of length one.
$3,5$ is an arithmetic progression of length two.
$3,5,7$ is an arithmetic progression of length three.
Then I took ...