Diophantine inequality involving binary forms

Abstract

Let d\ge \mathrm{3} be an integer, and set r={\mathrm{2}}^{d-\mathrm{1}}+\mathrm{1}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\mathrm{3}\le d\le \mathrm{4},r={\displaystyle \frac{\mathrm{17}}{\mathrm{32}}}\cdot {\mathrm{2}}^d+\mathrm{1}\hspace{0.17em}\mathrm{for}\hspace{0.17em}5\le d\le \mathrm{6},r=d^{\mathrm{2}}+d+\mathrm{1}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\mathrm{7}\le d\le \mathrm{8}, and r=d^{\mathrm{2}}+d+\mathrm{2}\hspace{0.17em}\mathrm{for }d\ge \mathrm{9}, respectively. Suppose that {{\displaystyle \Phi }}_i(x,y)\in \mathbb{Z}\left|x,y\right|(\mathrm{1}\le i\le r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂, . . . , λ_r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁ (x₁, y₁) + λ₂Φ₂ (x₂, y₂) + · · · + λ_rΦ_r (x_r, y_r) is indefinite. Then for any given real η and σ with 0<σ<22−d, it is proved that the inequality has infinitely many solutions in integers x₁, x₂, . . . , x_r, y₁, y₂, . . . , y_r. This result constitutes an improvement upon that of B. Q. Xue.

Keywords

Diophantine inequality / Davenport–Heilbronn method / binary form

References

[1] BourgainJ, DemeterC, GuthL. Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann of Math, 2016, 184(2): 633–682

[2] BrowningT D, DietmannR, P D T AElliott. Least zero of a cubic form. Math Ann, 2012, 352: 745–778

[3] CookR J. The value of additive forms at prime arguments. J Théor Nombres Bordeaux, 2001, 13: 77–91

[4] DavenportH, HeilbronnH. On indefinite quadratic forms in five variables. J Lond Math Soc, 1946, 21: 185–193

[5] TitchmarshE C. The Theory of the Riemann Zeta-Function. 2nd ed. Oxford: Oxford Univ Press, 1986

[6] VaughanR C. The Hardy-Littlewood method. Cambridge: Cambridge Univ Press, 1981

[7] WatsonG L. On indefinite quadratic forms in five variables. Proc Lond Math Soc, 1953, 3(3): 170–181

[8] Wooley T D. On Weyl’s inequality, Hua’s lemma, and exponential sums over binary forms. Duke Math J, 1999, 100: 373–423

[9] WooleyT D. Hua’s lemma and exponential sums over binary forms. In: Rational Points on Algebraic Varieties. Basel: Birkhäuser, 2001, 405–446

[10] WooleyT D. The cubic case of the main conjecture in Vinogradov’s mean value theorem. Adv Math, 2016, 294: 532–561

[11] XueB Q. Diophantine inequality involving binary forms. Front Math China, 2014, 9(3): 641–657

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany