A note on maximal progression-free sets

AbstractErdős et al [Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math. 200 (1999) 119–135.] asked whether there exists a maximal set of positive integers containing no three-term arithmetic progression and such that the difference of its adjacent elements approaches infinity. This note answers the question affirmatively by presenting such a set in which the difference of adjacent elements is strictly increasing. The construction generalizes to arithmetic progressions of any finite length

On Sums of Sets of Primes with Positive Relative Density

In this paper we show that if $A$ is a subset of the primes with positive relative density $\delta$, then $A+A$ must have positive upper density $C_1\delta e^{-C_2(\log(1/\delta))^{2/3}(\log\log(1/\delta))^{1/3}}$ in $\mathbb{N}$. Our argument applies the techniques developed by Green and Green-Tao used to find arithmetic progressions in the primes, in combination with a result on sums of subsets of the multiplicative subgroup of the integers modulo $M$.Comment: 21 pages, to appear in J. London Math. Soc., short remark added and typos fixe

Discrepancy of Sums of two Arithmetic Progressions

Estimating the discrepancy of the hypergraph of all arithmetic progressions in the set [N]=\{1,2,\hdots,N\} was one of the famous open problems in combinatorial discrepancy theory for a long time. An extension of this classical hypergraph is the hypergraph of sums of $k$ ($k\geq 1$ fixed) arithmetic progressions. The hyperedges of this hypergraph are of the form A_{1}+A_{2}+\hdots+A_{k} in $[N]$, where the $A_{i}$ are arithmetic progressions. For this hypergraph Hebbinghaus (2004) proved a lower bound of $\Omega(N^{k/(2k+2)})$. Note that the probabilistic method gives an upper bound of order $O((N\log N)^{1/2})$ for all fixed $k$. P\v{r}\'{i}v\v{e}tiv\'{y} improved the lower bound for all $k\geq 3$ to $\Omega(N^{1/2})$ in 2005. Thus, the case $k=2$ (hypergraph of sums of two arithmetic progressions) remained the only case with a large gap between the known upper and lower bound. We bridge his gap (up to a logarithmic factor) by proving a lower bound of order $\Omega(N^{1/2})$ for the discrepancy of the hypergraph of sums of two arithmetic progressions.Comment: 15 pages, 0 figure

Enumeration of three term arithmetic progressions in fixed density sets

Additive combinatorics is built around the famous theorem by Szemer\'edi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemer\'edi's theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a generalization of Szemer\'edi's theorem using methods from real algebraic geometry.Comment: 62 pages. Update v2: Corrected some references. Update v3: Incorporated feedbac