On differences and sums of integers, I

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

A set {b1,b2,...,bi} ⊂ {1,2,...,N} is said to be a difference intersector set if {a1,a2,...,as} ⊂ {1,2,...,N}, j > ε{lunate}N imply the solvability of the equation ax - ay = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b1,b2,...,bi} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations (ax - ay p = + 1, (au - av p) = - 1, (ar + as p) = + 1, (at + az p) = - 1 (where ( a p) denotes the Legendre symbol) and to show that "almost all" sets form both difference and sum intersector sets.

Original languageEnglish
Pages (from-to)430-450
Number of pages21
JournalJournal of Number Theory
Volume10
Issue number4
DOIs
Publication statusPublished - 1978

Fingerprint

Sum of integers
Sumsets
Solvability
Difference Set
Legendre
Theorem
Modulus
Denote
Imply

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

On differences and sums of integers, I. / Erdős, P.; Sárközy, A.

In: Journal of Number Theory, Vol. 10, No. 4, 1978, p. 430-450.

Research output: Contribution to journalArticle

@article{68cea860e5d84618bfc6e77437919c62,
title = "On differences and sums of integers, I",
abstract = "A set {b1,b2,...,bi} ⊂ {1,2,...,N} is said to be a difference intersector set if {a1,a2,...,as} ⊂ {1,2,...,N}, j > ε{lunate}N imply the solvability of the equation ax - ay = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b1,b2,...,bi} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations (ax - ay p = + 1, (au - av p) = - 1, (ar + as p) = + 1, (at + az p) = - 1 (where ( a p) denotes the Legendre symbol) and to show that {"}almost all{"} sets form both difference and sum intersector sets.",
author = "P. Erdős and A. S{\'a}rk{\"o}zy",
year = "1978",
doi = "10.1016/0022-314X(78)90017-3",
language = "English",
volume = "10",
pages = "430--450",
journal = "Journal of Number Theory",
issn = "0022-314X",
publisher = "Academic Press Inc.",
number = "4",

}

TY - JOUR

T1 - On differences and sums of integers, I

AU - Erdős, P.

AU - Sárközy, A.

PY - 1978

Y1 - 1978

N2 - A set {b1,b2,...,bi} ⊂ {1,2,...,N} is said to be a difference intersector set if {a1,a2,...,as} ⊂ {1,2,...,N}, j > ε{lunate}N imply the solvability of the equation ax - ay = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b1,b2,...,bi} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations (ax - ay p = + 1, (au - av p) = - 1, (ar + as p) = + 1, (at + az p) = - 1 (where ( a p) denotes the Legendre symbol) and to show that "almost all" sets form both difference and sum intersector sets.

AB - A set {b1,b2,...,bi} ⊂ {1,2,...,N} is said to be a difference intersector set if {a1,a2,...,as} ⊂ {1,2,...,N}, j > ε{lunate}N imply the solvability of the equation ax - ay = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b1,b2,...,bi} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations (ax - ay p = + 1, (au - av p) = - 1, (ar + as p) = + 1, (at + az p) = - 1 (where ( a p) denotes the Legendre symbol) and to show that "almost all" sets form both difference and sum intersector sets.

UR - http://www.scopus.com/inward/record.url?scp=28844449017&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=28844449017&partnerID=8YFLogxK

U2 - 10.1016/0022-314X(78)90017-3

DO - 10.1016/0022-314X(78)90017-3

M3 - Article

AN - SCOPUS:28844449017

VL - 10

SP - 430

EP - 450

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 4

ER -