Corpus ID: 119617623

Small Values of Indefinite Diagonal Quadratic Forms at Integer Points in at least five Variables

@article{Buterus2018SmallVO,
  title={Small Values of Indefinite Diagonal Quadratic Forms at Integer Points in at least five Variables},
  author={Paul Buterus and Friedrich Gotze and Thomas Hille},
  journal={arXiv: Number Theory},
  year={2018}
}
For any $\epsilon >0$ we derive an effective estimate for a solution of $|Q[m]| < \epsilon$ in non-zero integral points $m \in \mathbb Z^d \setminus \{0\}$ in terms of the signature $(r,s)$ and the largest eigenvalue, where $Q[x] = \sum_{i=1}^d \lambda_i x_i^2$ is a non-singular indefinite diagonal quadratic form of dimension $d \geq 5$. In order to prove our result, we extend an approach of Birch and Davenport(1958b) to higher dimensions combined with a theorem of Schlickewei (1985) on small… Expand
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References

SHOWING 1-10 OF 46 REFERENCES
Distribution of Values of Quadratic Forms at Integral Points
The number of lattice points in $d$-dimensional hyperbolic or elliptic shells $\{m;\; a<Q[m]<b\}$ which are restricted to rescaled and growing domains $r\4\Omega$ is approximated by the volume. AnExpand
Lattice point problems and values of quadratic forms
For d-dimensional ellipsoids E with d≥5 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order $\mathcal{O}(r^{d-2})$ forExpand
Indefinite quadratic forms in many variables
Let Q ( x 1 , …, x n ) be an indefinite quadratic form in n variables with real coefficients. It is conjectured that, provided n ≥ 5, the inequality is soluble for every e > 0 in integers x 1 , …, xExpand
Small zeros of quadratic forms
We give upper and lower bounds for zeros of quadratic forms in the rational, real and p-adic fields. For example, given r > 0, s > 0, there are infinitely many forms a with integer coefficients in rExpand
Quadratic forms of signature (2, 2) and eigenvalue spacings on rectangular 2-tori
The Oppenheim conjecture, proved by Margulis [Mar1] (see also [Mar2]), asserts that for a nondegenerate indefinite irrational quadratic form Q in n ≥ 3 variables, the set Q(Zn) is dense. In [EMM]Expand
Quadratic geometry of numbers
We give upper bounds for zeros of quadratic forms. For example we prove that for any nondegenerate quadratic form W(xl,...,xn) with rational integer coefficients which vanishes on a d-dimensionalExpand
Bounds for the least solutions of homogeneous quadratic equations
In this note I obtain bounds for the least integral solutions of the equationin terms ofFor ternary diagonal forms, such bounds have been given by Axel Thue (4) and, more recently, by Holzer (1),Expand
Lattice point problems and distribution of values of quadratic forms
For d-dimensional irrational ellipsoids E with d > 9 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order o(rd-2). TheExpand
Quadratic forms which have only large zeros
We give lower bounds for zeros of quadratic forms. For example, givenn≥2d>0 there are infinitely many nonsingular quadratic formsF with integral coefficients inn variables which vanish onExpand
Values of Random Polynomials at Integer Points
Using classical results of Rogers bounding the $L^2$-norm of Siegel transforms, we give bounds on the heights of approximate integral solutions of quadratic equations and error terms in theExpand
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