Almost-Arithmetic Progressions and Uniform Distribution

Arithmetic Progressions in Sumsets and L-almost-periodicity

We prove results about the L-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in L, and gives a very short proof of a theorem of Green that if A and B are subsets of {1, . . . , N} of sizes αN and βN then A + B contains an arithmetic progression of length at least exp ( c(αβ logN) − log logN ) . Another almost...

Almost disjoint families of 3-term arithmetic progressions

We obtain upper and lower bounds for the size of a largest family of 3-term arithmetic progressions contained in [0;n 1], no two of which intersect in more than one point. Such a family consists of just under a half of all the 3-term arithmetic progressions contained in [0;n 1]. MSC: 05D99

Constructing Small Sets that are Uniform in Arithmetic Progressions

On rainbow 4-term arithmetic progressions

{sl Let $[n]={1,dots, n}$ be colored in $k$ colors. A rainbow AP$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. Conlon, Jungi'{c} and Radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow AP(4) free, when $n$ is even. Based on their construction, we show that such a coloring of $[4n]$...

Primes in arithmetic progressions

Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli k ≤ 72 and other small moduli.