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A survey and a result on inhomogeneous quadratic forms

Sourav Das, Anish Ghosh

TL;DR

This work investigates the values of inhomogeneous quadratic forms $Q_{\boldsymbol\xi}(y)=Q(y+\boldsymbol\xi)$ at integer points, connecting Oppenheim-type questions to homogeneous dynamics on spaces of affine lattices. The authors develop a unified dynamical framework, employing Schmidt games and reductions to discrete-time actions to obtain thick/dimension-type results, including a full Hausdorff-dimension statement for two-variable forms whose values avoid a fixed countable set not containing $0$. They also present effective and quantitative results across several averaging regimes, including fixed rational shifts, averaging over smaller families, and $S$-adic analogues, situating these in the broader context of ergodic theory for semisimple group actions. The methods hinge on the AGK program and KW’s binary-form analysis, offering a versatile toolkit that bridges Diophantine approximation, building on Margulis–Dani–Ratner theory, with potential extensions to more general inhomogeneous problems. Overall, the paper reinforces the deep link between Oppenheim-type questions and homogeneous dynamics, while providing new thickness/dimension results and a coherent strategy for effectivity across various settings.

Abstract

We survey recent work done on the values at integer points of irrational inhomogeneous quadratic forms, namely, inhomogeneous analogues of the famous Oppenheim conjecture. We also prove that the set of such forms in two variables whose set of values at integer points avoids a given countable set not containing zero, has full Hausdorff dimension. Moreover, we consider the more refined variant of this problem, where the shift is fixed and the form is allowed to vary. The strategy is to translate the problems to homogeneous dynamics and deduce the theorems from their dynamical counterparts. While our approach is inspired by the work of Kleinbock and Weiss, the dynamical results can be deduced from more general results of An, Guan, and Kleinbock.

A survey and a result on inhomogeneous quadratic forms

TL;DR

This work investigates the values of inhomogeneous quadratic forms at integer points, connecting Oppenheim-type questions to homogeneous dynamics on spaces of affine lattices. The authors develop a unified dynamical framework, employing Schmidt games and reductions to discrete-time actions to obtain thick/dimension-type results, including a full Hausdorff-dimension statement for two-variable forms whose values avoid a fixed countable set not containing . They also present effective and quantitative results across several averaging regimes, including fixed rational shifts, averaging over smaller families, and -adic analogues, situating these in the broader context of ergodic theory for semisimple group actions. The methods hinge on the AGK program and KW’s binary-form analysis, offering a versatile toolkit that bridges Diophantine approximation, building on Margulis–Dani–Ratner theory, with potential extensions to more general inhomogeneous problems. Overall, the paper reinforces the deep link between Oppenheim-type questions and homogeneous dynamics, while providing new thickness/dimension results and a coherent strategy for effectivity across various settings.

Abstract

We survey recent work done on the values at integer points of irrational inhomogeneous quadratic forms, namely, inhomogeneous analogues of the famous Oppenheim conjecture. We also prove that the set of such forms in two variables whose set of values at integer points avoids a given countable set not containing zero, has full Hausdorff dimension. Moreover, we consider the more refined variant of this problem, where the shift is fixed and the form is allowed to vary. The strategy is to translate the problems to homogeneous dynamics and deduce the theorems from their dynamical counterparts. While our approach is inspired by the work of Kleinbock and Weiss, the dynamical results can be deduced from more general results of An, Guan, and Kleinbock.
Paper Structure (9 sections, 23 theorems, 89 equations)

This paper contains 9 sections, 23 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Questions of effectivity
  3. Fixed rational shifts
  4. Uniformity and Averaging along smaller families
  5. Values of quadratic forms at S integers
  6. Binary inhomogeneous forms
  7. Schmidt Games
  8. Hyperplane percentage game
  9. Reduction to discrete time action

Key Result

Theorem 1.1

For any indefinite, irrational and nondegenerate inhomogeneous quadratic form $Q_{\xi}$ in $n \geq 3$ variables, there is $c_Q>0$ such that while for $n\geq 5$ the limit exists and equals $c_Q|I|$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 3.1
  • Theorem 3.2
  • Definition 5.1
  • ...and 25 more