Quadratic Diophantine Inequalities

Abstract

We consider systems of quadratic diophantine inequlities. For example, suppose that Q 1 and Q 2 are real diagonal quadratic forms in s variables, where one has s 10. Suppose also that every form :Q 1 +;Q 2 with (:, ;) # R 2 "[0] has at least 5 nonzero coefficients, one irrational coefficient, at least one negative coefficient, and at least one positive coefficient. Then for any =>0, there exists a nonzero integral vector x # Z s such that |Q 1 (x)| <= and |Q 2 (x)| <=. We also prove a result on systems of R quadratic diophantine inequalities under more complicated restrictions. Academic Press

Asymptotic lower bounds for Diophantine inequalities

“…Finally, we make a few comments about systems of R diagonal Diophantine inequalities of odd degree. It seems very likely that some sort of result analogous to the results of this paper would hold, by combining the methods in this paper with methods of Briidern and Cook [5] and Freeman [12]. Suppose that F l ,F 2 ... ,F R are the relevant diagonal forms in s variables.…”

Systems of cubic Diophantine inequalities

“…The result follows in exactly the same manner as Lemma 4.4 of [17]. In fact, what is essentially needed, for all of the above statements, is the condition l f 7, which certainly follows from (12). The absolute convergence of the singular series follows from the Corollary to Lemma 2.10 of [16].…”

“…Although they only explicitly consider forms of degree k f 13 and thus do not consider cubic forms, this does not alter the treatment substantially at all. Finally, for more details in a proof of a result similar to Lemma 2, see [12], Section 3.4.1. In fact, what is essentially needed, for all of the above statements, is the condition l f 7, which certainly follows from (12).…”

“…Assume that R is a positive integer satisfying R f 2, that r is an integer with 0 e r e R, and that l is defined as in (12). Assume that R is a positive integer satisfying R f 2, that r is an integer with 0 e r e R, and that l is defined as in (12).…”

“…For an alternate (and much longer) approach using Fourier integral's theorem, see [12], section 3.4.3. For an alternate (and much longer) approach using Fourier integral's theorem, see [12], section 3.4.3.…”