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Solving quadratic forms in restricted variables with the circle method

Mieke Wessel, Svenja zur Verth

TL;DR

This work develops a circle-method framework to count weighted integral solutions of a non-singular quadratic form f(\mathbf{x})=t with bounded height, under minimal structural assumptions on the form and the weight sequence \mathcal{A}. By imposing a rank condition on a submatrix of f (Condition L_1) and distribution and regularity conditions on the weights (Conditions L_2, CC, CD, CK, CL1, CL2), the authors derive an asymptotic formula R_{f,t}(X,\mathcal{A})=\mathfrak{S}\mathfrak{I}(X)+O( errors ), where the main term factors into a product of a p-adic singular series and a real singular integral. The singular series is interpreted as a product of weighted p-adic densities and the singular integral as a real density of solutions within a weighted, smoothly-approximated measure; these densities yield a local-global picture for the counting problem. The paper also treats a box-restricted version R_{f,t}(X,\mathcal{A},\mathfrak{B}) and provides concrete examples, including prime and k-free weights, illustrating how the framework recovers and extends known results in the literature.

Abstract

Let $f(\mathbf x)$ be a non-singular quadratic form with sufficiently many mixed terms and $t$ an integer. For a sequence of weights $\mathcal A$ we study the number of weighted solutions to $f(\mathbf x) = t$. In particular, we give conditions on both $\mathcal A$ and $f$ such that we can use the circle method to count such solutions of bounded height.

Solving quadratic forms in restricted variables with the circle method

TL;DR

This work develops a circle-method framework to count weighted integral solutions of a non-singular quadratic form f(\mathbf{x})=t with bounded height, under minimal structural assumptions on the form and the weight sequence \mathcal{A}. By imposing a rank condition on a submatrix of f (Condition L_1) and distribution and regularity conditions on the weights (Conditions L_2, CC, CD, CK, CL1, CL2), the authors derive an asymptotic formula R_{f,t}(X,\mathcal{A})=\mathfrak{S}\mathfrak{I}(X)+O( errors ), where the main term factors into a product of a p-adic singular series and a real singular integral. The singular series is interpreted as a product of weighted p-adic densities and the singular integral as a real density of solutions within a weighted, smoothly-approximated measure; these densities yield a local-global picture for the counting problem. The paper also treats a box-restricted version R_{f,t}(X,\mathcal{A},\mathfrak{B}) and provides concrete examples, including prime and k-free weights, illustrating how the framework recovers and extends known results in the literature.

Abstract

Let be a non-singular quadratic form with sufficiently many mixed terms and an integer. For a sequence of weights we study the number of weighted solutions to . In particular, we give conditions on both and such that we can use the circle method to count such solutions of bounded height.

Paper Structure

This paper contains 8 sections, 20 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Set-up and notation
  3. A Weyl-like estimate and the minor arcs
  4. Treatment of the major arcs
  5. Interpretation of the singular series
  6. Interpretation of the singular integral
  7. Proof of the main theorems
  8. Examples

Key Result

Theorem 1.6

Assume Conditions CC, CD, CK, CL1 and CL2 hold and that $A(X)$ is smoothly approximable (see Definition Dsmthaprox below). Furthermore, let $K>0$ be any real number and define $\mathfrak I(X)$ as in Equation (EQ truncated sing series). Then Here $\mathfrak S$ is a non-negative constant that can be interpreted as the product of local $p$-adic densities for all primes $p$.

Theorems & Definitions (54)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Definition 2.1
  • Example 2.2
  • ...and 44 more