Arithmetic Progressions with Restricted Digits

2018/09/07 by Walker, Aled, Walker, Alexander
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1809.02430

Abstract

For an integer b \geqslant 2 and a set S⊂ \0,⋯,b-1\, we define the Kempner set K(S,b) to be the set of all non-negative 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 paper 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.

Related

• The number of subsets of integers with no k-term arithmetic progression
• On sets of integers not containing long arithmetic progressions
• Numbers omitting digits in certain base expansions
• Digitally Restricted Sets and the Goldbach Conjecture: An Exceptional Set Result
• Finding arithmetic progressions in dense sets of integers