## Doctoral Theses

### Sequences of Positive Integers Containing no K-Term Arithmetic Progressions and Smooth Numbers in Short Intervals.

2-28-2009

2-28-2010

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)

#### Abstract (Summary of the Work)

In my thesis I have worked on two problems:1. On sequences of positive integers containing no k terms in arithmetic progressions.2. On smooth numbers in short intervals.The first two chapters of my thesis deal with the first problem and in the rest of the thesis I have focused on the 2nd problem.In the first chapter of my thesis I have considered the function rk(N) for a fixed k â‰¥ 3, where, by definition, rk(N) is the cardinality of a maximal subset of N consecutive natural numbers with the property that nork terms of it are in an Arithmetic Progression (A. P.). Obtaining â€˜goodâ€™ estimates for rk(N) for sufficiently large N is one of the most challenging problems in this area. In section 2 of this chapter, I have listed some results on the lower and upper bounds of rk(N). Rankinâ€™s theorem is the best known result on the lower bound of rk(N) and SzemerÂ´ediâ€™s theorem is the best known result on the upper bound of rk(N). I have presented proofs of many known results on lower bounds of rk(N) in section 3. In particular I have presented a result which is a little weaker than Rankinâ€™s theorem for k = 3. I have used this result in chapter 2 to work on our problem. In section 4 of this chapter, I have presented the proofs of many known results on the upper bounds of rk(N). In particular, I have sketched the proof of SzemerÂ´ediâ€™s theorem for k = 3. I could not access the proof of theorem 1.4.4 of chapter 1. The proof that I have given is my own.SzemerÂ´ediâ€™s theorem is a consequence of ErdÂ¨os conjecture on sequences of positive integers containing no k terms in an A. P. I have shown this fact in Corollary 2.2.6 of chapter 2. In Chapter 2, I have assumed the conjecture of ErdÂ¨os and obtained a very strong consequence from it.This famous conjecture of ErdÂ¨os asserts that if A is a subset of the set of all ositive integers having the property that âˆ‘aâˆˆA 1/a = âˆž, then A must contain arithmetic progressions of arbitrary length. A special case of the conjecture, when A is the set of prime numbers, was proved by Green and Tao [GT08]. The conjecture implies that if a subset A of the set of positive integers contains no arithmetic progression of length k, where k â‰¥ 3 is a fixed integer, then the sum âˆ‘aâˆˆA 1/a must converge. One may ask whether the above sum can be arbitrarily large as the sets A vary. Our first theorem of this chapter answers the question in the negative.Joseph L. Gerver [Ger77] considered the set Sp, given by the sequence {an}, where a1 = 1 and for n â‰¥ 1, an+1 is the smallest positive integer bigger than an such that no p elements of a1, a2, Â· Â· Â· , an+1 lie in arithmetic progression.

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