## Algebra & Number Theory

### Equidistribution of values of linear forms on a cubic hypersurface

Sam Chow

#### 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 $L$ at integer zeros of $C$ are equidistributed modulo $1$ in $ℝr$, requiring only that $h > 16$.

#### Article information

Source
Algebra Number Theory, Volume 10, Number 2 (2016), 421-450.

Dates
Revised: 9 November 2015
Accepted: 27 December 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842483

Digital Object Identifier
doi:10.2140/ant.2016.10.421

Mathematical Reviews number (MathSciNet)
MR3477746

Zentralblatt MATH identifier
06561469

#### Citation

Chow, Sam. Equidistribution of values of linear forms on a cubic hypersurface. Algebra Number Theory 10 (2016), no. 2, 421--450. doi:10.2140/ant.2016.10.421. https://projecteuclid.org/euclid.ant/1510842483

#### References

• V. Bentkus and F. G ötze, “Lattice point problems and distribution of values of quadratic forms”, Ann. of Math. $(2)$ 150:3 (1999), 977–1027.
• B. J. Birch, “Forms in many variables”, Proc. Roy. Soc. $($A$)$ 265 (1962), 245–263.
• T. D. Browning, Quantitative arithmetic of projective varieties, Progress in Mathematics 277, Birkhäuser, Basel, 2009.
• T. D. Browning, R. Dietmann, and D. R. Heath-Brown, “Rational points on intersections of cubic and quadric hypersurfaces”, J. Inst. Math. Jussieu 14:4 (2015), 703–749.
• J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics 45, Cambridge University, New York, 1957.
• S. G. Dani and G. A. Margulis, “Limit distributions of orbits of unipotent flows and values of quadratic forms”, pp. 91–137 in I. M. Gelfand Seminar, edited by S. Gelfand and S. Gindikin, Advances in Soviet Mathematics 16, American Mathematical Society, Providence, RI, 1993.
• H. Davenport, “Cubic forms in thirty-two variables”, Philos. Trans. Roy. Soc. London. $($A$)$ 251:993 (1959), 193–232.
• H. Davenport, Analytic methods for Diophantine equations and Diophantine inequalities, 2nd ed., Cambridge University, 2005.
• H. Davenport and H. Heilbronn, “On indefinite quadratic forms in five variables”, J. London Math. Soc. 21:3 (1946), 185–193.
• H. Davenport and D. J. Lewis, “Non-homogeneous cubic equations”, J. London Math. Soc. 39 (1964), 657–671.
• A. Eskin, G. A. Margulis, and S. Mozes, “Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture”, Ann. of Math. $(2)$ 147:1 (1998), 93–141.
• D. E. Freeman, “Asymptotic lower bounds and formulas for Diophantine inequalities”, pp. 57–74 in Number theory for the millennium, II (Urbana, IL, 2000), edited by M. A. Bennett et al., A K Peters, Natick, MA, 2002.
• S. W. Graham and G. Kolesnik, Van der Corput's method of exponential sums, London Mathematical Society Lecture Note Series 126, Cambridge University, 1991.
• D. R. Heath-Brown, “Cubic forms in ten variables”, Proc. London Math. Soc. $(3)$ 47:2 (1983), 225–257.
• D. R. Heath-Brown, “A new form of the circle method, and its application to quadratic forms”, J. Reine Angew. Math. 481 (1996), 149–206.
• D. R. Heath-Brown, “Cubic forms in $14$ variables”, Invent. Math. 170:1 (2007), 199–230.
• G. A. Margulis, “Discrete subgroups and ergodic theory”, pp. 377–398 in Number theory, trace formulas and discrete groups (Oslo, 1987), edited by K. E. Aubert et al., Academic Press, Boston, 1989.
• O. Sargent, “Equidistribution of values of linear forms on quadratic surfaces”, Algebra Number Theory 8:4 (2014), 895–932.
• W. M. Schmidt, “On cubic polynomials, IV: Systems of rational equations”, Monatsh. Math. 93:4 (1982), 329–348.
• W. M. Schmidt, “Simultaneous rational zeros of quadratic forms”, pp. 281–307 in Séminaire de Théorie des Nombres: Séminaire Delange–Pisot–Poitou (Paris, 1980–1981), edited by M.-J. Bertin, Progress in Mathematics 22, Birkhäuser, Boston, 1982.
• W. M. Schmidt, “The density of integer points on homogeneous varieties”, Acta Math. 154:3–4 (1985), 243–296.
• R. C. Vaughan, The Hardy–Littlewood method, 2nd ed., Cambridge Tracts in Mathematics 125, Cambridge University, 1997.
• T. D. Wooley, “On Diophantine inequalities: Freeman's asymptotic formulae”, in Proceedings of the Session in Analytic Number Theory and Diophantine Equations (Bonn, 2002), edited by D. R. Heath-Brown and B. Z. Moroz, Bonner Mathematische Schriften 360, Universität Bonn, 2003.