Monochromatic arithmetic progressions in binary Thue–Morse-like words

نویسندگان

چکیده

We present results on the existence of long arithmetic progressions in Thue-Morse word and a class generalised words. Our arguments are inspired by van der Waerden's proof for arbitrary monochromatic any finite colouring (positive) integers.

برای دانلود باید عضویت طلایی داشته باشید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

A probabilistic threshold for monochromatic arithmetic progressions

We show that √ k2k/2 is, roughly, the threshold where, under mild conditions, on one side almost every coloring contains a monochromatic k-term arithmetic progression, while on the other side, there are almost no such colorings.

متن کامل

Monochromatic Arithmetic Progressions With Large Differences

A generalisation of the van der Waerden numbers w(k;r) is considered. For a function f : Z! R + define w(k; f ;r) to be the least positive integer (if it exists) such that for every r-coloring of [1;w( f ;k;r)] there is a monochromatic arithmetic progression fa+ id : 0 i k 1g such that d f (a). Upper and lower bounds are given for w( f ;3;2). For k > 3 or r > 2, particular functions f are given...

متن کامل

Words avoiding repetitions in arithmetic progressions

Carpi constructed an infinite word over a 4-letter alphabet that avoids squares in all subsequences indexed by arithmetic progressions of odd difference. We show a connection between Carpi’s construction and the paperfolding words. We extend Carpi’s result by constructing uncountably many words that avoid squares in arithmetic progressions of odd difference. We also construct infinite words avo...

متن کامل

Monochromatic 4-term arithmetic progressions in 2-colorings of Zn

This paper is motivated by a recent result of Wolf [12] on the minimum number of monochromatic 4-term arithmetic progressions (4-APs, for short) in Zp, where p is a prime number. Wolf proved that there is a 2-coloring of Zp with 0.000386% fewer monochromatic 4-APs than random 2-colorings; the proof is probabilistic and non-constructive. In this paper, we present an explicit and simple construct...

متن کامل

Arithmetic Progressions and Binary Quadratic Forms

is a (nonconstant) arithmetic progression of positive integers. We consider a general binary quadratic form ax2 + bxy + cy' ( a , b , c E Z ) and ask the question "Can the form ax' + hxy + ry' represen1 every inleger in 1he arithmetic progression kNo + 1 for any natural numbers k and l?" In a sampling of books containing a discussion of binary quadratic forms [2]-[9], we did not find this qustl...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

ژورنال

عنوان ژورنال: Theoretical Computer Science

سال: 2022

ISSN: ['1879-2294', '0304-3975']

DOI: https://doi.org/10.1016/j.tcs.2022.08.013