Discrepancy in arithmetic progressions

It is proven that there is a two-coloring of the first integers for which all arithmetic progressions have discrepancy less than . This shows that a 1964 result of K. F. Roth is, up to constants, best possible.

     

On some Rado numbers for generalized arithmetic progressions

The 2-color Rado number for the equation x₁+x₂−2x₃=c, which for each constant we denote by S₁(c), is the least integer, if it exists, such that every 2-coloring, Δ : [1,S₁(c)]→{0,1}, of the natural numbers admits a monochromatic solution to x₁+x₂−2x₃=c, and otherwise S₁(c)=∞. We determine the 2-color Rado number for the equation x₁+x₂−2x₃=c, when additional inequality restraints on the variables are added. In particular, the case where we require x₂<x₃<x₁, is a generalization of the 3-term arithmetic progression; and the work done here improves previously established upper bounds to an exact value.     

Powerful arithmetic progressions

We give a complete characterization of so-called powerful arithmetic progressions, i.e. of progressions whose kth term is a kth power for all k. We also prove that the length of any primitive arithmetic progression of powers can be bounded both by any term of the progression different from 0 and ±1, and by its common difference. In particular, such a progression can have only finite length.     

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 and other small moduli.

     

Near arithmetic progressions in sparse sets

Nonstandard methods are used to obtain results in combinatorial number theory. The main technique is to use the standard part map to translate density properties of subsets of into Lebesgue measure properties on . This allows us to obtain a simple condition on a standard sequence that guarantees the existence of intervals in arithmetic progression, all of which contain elements of with various uniform density conditions.

     

Rankin-Selberg coefficients in large arithmetic progressions

Let (λ_f(n))_n≥1 be the Hecke eigenvalues of either a holomorphic Hecke eigencuspform or a Hecke-Maass cusp form f.We prove that,for any fixed η> 0,under the Ramanujan-Petersson conjecture for GL₂ Maass forms,the Rankin-Selberg coefficients(λ_f(n)²)_n≥1 admit a level of distribution θ=2/5+1/260-η in arithmetic progressions.     

On infinite arithmetic progressions in sumsets

Let k be a positive integer.Denote by D_1/k the least integer d such that for every set A of nonnegative integers with the lower density 1/k,the set(k+1)A contains an infinite arithmetic progression with difference at most d,where(k+1)A is the set of all sums of k+1 elements(not necessarily distinct) of A.Chen and Li(2019) conjectured that D_1/k=k²+o(k²).The purpose of this paper is to confirm the above conjecture.We also prove that D_1/k is a ...     

Long arithmetic progressions in critical sets

Given a density 0<σ?1, we show for all sufficiently large primes p that if S⊆Z/pZ has the least number of three-term arithmetic progressions among all sets with at least σp elements, then S contains an arithmetic progression of length at least log^1/4+o(1)p.     

Convergent sieve sequences in arithmetic progressions

Let V be a set of pairwise coprime integers not containing 1 and suppose, there is a 0?δ<1, such that _∑v∈Vv−1+δ<∞ holds. Let χV(n)=1 if v?n for all v∈V and χV(n)=0 elsewhere. We study the behavior of χV in arithmetic progressions uniformly in the modulus, both individually and in the quadratic mean over the residue classes. As an application, new bounds for the mean square error of squarefree numbers in arithmetic progressions are obtained.     

Sub-Ramsey numbers for arithmetic progressions

Letm 3 andk 1 be two given integers. Asub-k-coloring of [n] = {1, 2,...,n} is an assignment of colors to the numbers of [n] in which each color is used at mostk times. Call an arainbow set if no two of its elements have the same color. Thesub-k-Ramsey number sr(m, k) is defined as the minimumn such that every sub-k-coloring of [n] contains a rainbow arithmetic progression ofm terms. We prove that((k – 1)m ²/logmk) sr(m, k) O((k – 1)m ² logmk) asm , and apply the same method to improve a previously known upper bound for a problem concerning mappings from [n] to [n] without fixed points.Research supported in part by Allon Fellowship and by a Bat Sheva de-Rothschild grant.Research supported in part by the AKA Research Fund of the Hungarian Academy of Sciences, grant No. 1-3-86-264.     

A remark concerning arithmetic progressions

F. Cohen raised the following question: Determine or estimate a function F(d) so that if we split the integers into two classes at least one class contains, for infinitely many values of d, an arithmetic progression of difference d and length F(d). We prove F(d) ? (1 + ε) log₂d.