• Corpus ID: 248863451

Some new results in quantitative Diophantine approximation

  title={Some new results in quantitative Diophantine approximation},
  author={Anish Ghosh and V. Vinay Kumaraswamy},
. In this article we establish two new results on quantitative Diophantine approximation for one-parameter families of diagonal ternary indefinite forms. In the first result, we consider quadratic forms taking values at prime points. In the second, we examine inhomogeneous forms of arbitrary degree taking values at integer points. 


Diophantine Inequalities for Generic Ternary Diagonal Forms
  • D. Schindler
  • Mathematics
    International Mathematics Research Notices
  • 2018
Let $k\geq 2$ and consider the Diophantine inequality $$ \left|x_1^k-{\alpha}_2 x_2^k-{\alpha}_3 x_3^k\right| <{\theta}.$$Our goal is to find non-trivial solutions in the variables $x_i$, $1\leq
A Quantitative Oppenheim Theorem for generic ternary quadratic forms
We prove a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms. Our results are inspired by and analogous to recent results for diagonal quadratic forms due
Distribution of values of quadratic forms at integral points
<jats:p>The number of lattice points in <jats:italic>d</jats:italic>-dimensional hyperbolic or elliptic shells <jats:inline-formula><jats:alternatives><jats:tex-math>$$\{m :
Effective estimates on indefinite ternary forms
We give an effective proof of a theorem of Dani and Margulis regarding values of indefinite ternary quadratic forms at primitive integer vectors. The proof uses an effective density-type result for
Diophantine Approximation by Prime Numbers
On indefinite quadratic forms in four variables
On the difference between consecutive primes
Update: This work reproduces an earlier result of Peck, which the author was initially unaware of. The method of the proof is essentially the same as the original work of Peck. There are no new
On some diophantine inequalities involving primes.
On a theorem of Davenport, and Heilbronn
Values of inhomogeneous forms at S‐integral points
We prove effective versions of Oppenheim's conjecture for generic inhomogeneous forms in the S‐arithmetic setting. We prove an effective result for fixed rational shifts and generic forms and we also