# An analytic approach to some diophantine inequalities

### Gajraj, J.

#### (1976)

Gajraj, J. (1976) An analytic approach to some diophantine inequalities.

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#### Abstract

Heilbronn proved that for any epsilon > 0 there exists a number C (epsilon) such that for any real numbers theta and N(≥ 1) min n

2 theta-1/2+epsilon In the first part of this thesis we prove various extensionsof this result. We find values of g(r, k, s) so that the inequality [equation] is soluble, where X is an integral s-dimensional vector and the f's are either polynomials (without constant term) or forms, both of degree ≤1 k. The method used depends upon estimates for certain exponential sums. Using Weyl's estimates, we look, in Chapter 2, at monomials of different degree, and, in Chapter 3, at additive forms of degree k in s variables and quadratic polynomials. Using Hua's improvement of Vinogradov's estimates, we improve, for large values of k, the results of Chapter 2 and the results of Chapter 3 on additive forms. Also using Hua's estimates, we look, in Chapter 6 at polynomials of any degree (≤ k). In the course of this work we improve some results of Liu and Cook. Birch, Davenport and Ridout proved that if Q is an indefinite quadratic form in n( ≥ 21) variables, the inequality |Q (X)| < epsilon, has an integral solution X with |x| ≥. In the second part of the thesis we investigate the inequality [equation] where Q is an indefinite quadratic form in n (≥21) variables of rank r, and k = min (r, n-r). We find values for f(n, k) and show that (i) for k ≥ 6, lim(N->infinity) f(n, k) = 1/2, (ii) for k ≥ 7, 10k 3n lim f(n, k) = ½, and (iii) for 10k ≥ 3n, lim(N->infinity) f(n, k) = 1/2, k->infinity

This is a Accepted version
This version's date is: 1976
This item is not peer reviewed

Item TypeThesis (Doctoral)
TitleAn analytic approach to some diophantine inequalities
AuthorsGajraj, J.
Uncontrolled KeywordsMathematics; Pure Sciences; Analytic; Approach; Diophantine; Diophantine Inequalities; Diophantine Inequalities; Inequalities; Some
Departments

#### Identifiers

ISBN978-1-339-61921-7

Deposited by () on 01-Feb-2017 in Royal Holloway Research Online.Last modified on 01-Feb-2017

#### Notes

Digitised in partnership with ProQuest, 2015-2016. Institution: University of London, Royal Holloway College (United Kingdom).