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

Discrepancy in generalized arithmetic progressions

AbstractEstimating the discrepancy of the set of all arithmetic progressions in the first N natural numbers was one of the famous open problems in combinatorial discrepancy theory for a long time, successfully solved by K. Roth (lower bound) and Beck (upper bound). They proved that D(N)=minχmaxA|∑x∈Aχ(x)|=Θ(N1/4), where the minimum is taken over all colorings χ:[N]→{−1,1} and the maximum over all arithmetic progressions in [N]={0,…,N−1}.Sumsets of k arithmetic progressions, A1+⋯+Ak, are called k-arithmetic progressions and they are important objects in additive combinatorics. We define Dk(N) as the discrepancy of the set {P∩[N]:P is a k-arithmetic progression}. The second author proved that Dk(N)=Ω(Nk/(2k+2)) and Přívětivý improved it to Ω(N1/2) for all k≥3. Since the probabilistic argument gives Dk(N)=O((NlogN)1/2) for all fixed k, the case k=2 remained the only case with a large gap between the known upper and lower bounds. We bridge this gap (up to a logarithmic factor) by proving that Dk(N)=Ω(N1/2) for all k≥2.Indeed we prove the multicolor version of this result

Discrepancy of arithmetic structures

In discrepancy theory, we investigate how well a desired aim can be achieved. So typically we do not compare our solution with an optimal solution, but rather with an (idealized) aim. For example, in the declustering problem, we try to distribute data on parallel disks in such a way that all of a prespecified set of requests find their data evenly distributed on the disks. Hence our (idealized) aim is that each request asks for the same amount of data from each disk. Structural results tell us to which extent this is possible. They determine the discrepancy, the deviation of an optimal solution from our aim. Algorithmic results provide good declustering scheme. We show that for grid structure data and rectangle queries, a discrepancy of order (log M)^((d-1)/2) cannot be avoided. Moreover, we present a declustering scheme with a discrepancy of order (log M)^(d-1). Furthermore, we present discrepancy results for hypergraphs related to the hypergraph of arithmetic progressions, for the hypergraph of linear hyperplanes in finite vector spaces and for products of hypergraphs