Arithmetic Progressions with Restricted Digits. Issue 2 (7th February 2020)

• Record Type: Journal Article
• Authors: Walker, Aled; Walker, Alexander
• Is Part Of: American Mathematical Monthly. Volume 127:Issue 2(2020)
• Page Start: 140
• Page End: 150
• Publication Date: 2020-02-07
• Subjects: Primary 11B25 -- Secondary 11N35; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2020.1682888
• Languages: English
• ISSNs: 0002-9890

Abstract

Abstract: For an integer b ⩾ 2 and a set S ⊂ { 0, …, b − 1 }, we define the Kempner set K ( S, b ) to be the set of all nonnegative integers whose base- b digital expansions contain only digits from S . These well-studied sparse sets provide a rich setting for additive number theory, and in this article we study various questions relating to the appearance of arithmetic progressions in these sets. In particular, for all b we determine exactly the maximal length of an arithmetic progression that omits a base- b digit.