Equidistribution of values of linear forms on a cubic hypersurface

Sam Chow

Research output: Contribution to journalArticle (Academic Journal)peer-review

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Abstract

Let C be a cubic form with integer coefficients in n variables, and let h be the h-invariant of C. Let L1,…,Lr be linear forms with real coefficients such that, if α∈ℝr∖{0}, then α⋅L is not a rational form. Assume that h>16+8r. Let τ∈ℝr, and let η be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions x∈[−P,P]n to the system C(x)=0, |L(x)−τ|<η. If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the h-invariant condition with the hypothesis n>16+9r and show that the system has an integer solution. Finally, we show that the values of LL at integer zeros of C are equidistributed modulo 1 in ℝr, requiring only that h>16.
Original languageEnglish
Pages (from-to)421-450
Number of pages30
JournalAlgebra and Number Theory
Volume10
Issue number2
DOIs
Publication statusPublished - 16 Mar 2016

Keywords

  • diophantine equations
  • diophantine inequalities
  • diophantine approximation
  • equidistribution

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