LARGE SUBSETS OF EUCLIDEAN SPACE AVOIDING INFINITE ARITHMETIC PROGRESSIONS

被引：5

作者：

Bradford, Laurestine ^{[1
,2
]}

Kohut, Hannah ^{[3
]}

Mooroogen, Yuveshen ^{[3
]}

机构：

[1] McGill Univ, Dept Linguist, Montreal, PQ H3A 1A7, Canada

[2] Ctr Res Brain Language & Mus, Montreal, PQ H3G 2A8, Canada

[3] Univ British Columbia, Dept Math, Vancouver, BC V6T 1Z2, Canada

来源：

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY | 2023年 / 151卷 / 08期

基金：

加拿大自然科学与工程研究理事会;

关键词：

SETS;

D O I：

10.1090/proc/16404

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

It is known that if a subset of R has positive Lebesgue measure, then it contains arbitrarily long finite arithmetic progressions. We prove that this result does not extend to infinite arithmetic progressions in the following sense: for each lambda in [0, 1), we construct a subset of R that intersects every interval of unit length in a set of measure at least lambda, but that does not contain any infinite arithmetic progression.

引用

页码：3535 / 3545

页数：11

共 16 条

[1] ON SETS OF INTEGERS WHICH CONTAIN NO 3 TERMS IN ARITHMETICAL PROGRESSION [J].

BEHREND, FA .

PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, 1946, 32 (12) :331-332

[2]

Burgin Alex, 2022, PREPRINT

[3]

Chlebik Miroslav, 2015, PREPRINT

[4]

Denson J., 2021, Springer Optim. Appl., V168, P59, DOI [10.1007/978-3-030-61887-2_4, DOI 10.1007/978-3-030-61887-2_4]

[5] LARGE SETS AVOIDING PATTERNS [J].

Fraser, Robert ;

Pramanik, Malabika .

ANALYSIS & PDE, 2018, 11 (05) :1083-1111

[6] CONSTRUCTION OF ONE-DIMENSIONAL SUBSETS OF THE REALS NOT CONTAINING SIMILAR COPIES OF GIVEN PATTERNS [J].

Keleti, Tamas .

ANALYSIS & PDE, 2008, 1 (01) :29-33

[7]

Kolountzakis Mihail N., 2022, PREPRINT

[8]

Kuhl J., 2022, Personal communication

[9]

Maga P, 2010, REAL ANAL EXCH, V36, P79

[10] Sets of large dimension not containing polynomial configurations [J].

Mathe, Andras .

ADVANCES IN MATHEMATICS, 2017, 316 :691-709

← 1 2 →