Diophantine Approximation with a Quaternary Problem

Abstract

Let $1 < k < {7 \over 6},{\lambda _1},{\lambda _2},{\lambda _3}$ and λ₄ be non-zero real numbers, not all of the same sign such that ${{{\lambda _1}} \over {{\lambda _2}}}$ is irrational and let ω be a real number. We prove that the inequality $\left| {{\lambda _1}p_1^2 + {\lambda _2}p_2^2 + {\lambda _3}p_3^2 + {\lambda _4}p_4^k - \omega} \right| \le {\left({{{\max}_j}{p_j}} \right)^{- {{7 - 6k} \over {14k}} + \varepsilon}}$ has infinitely many solutions in prime variables p₁, p₂, p₃, p₄ for any ε > 0.

Keywords

Diophantine inequalities / Goldbach-type problem / Hardy–Littlewood method / 11D75 / 11J25 / 11P32 / 11P55

References

[1] BakerR, HarmanG. Diophantine approximation by prime numbers. J. London Math. Soc. (2), 1982, 25(2): 201-215.

[2] BrüdernJ, CookR, PerelliA. The values of binary linear forms at prime arguments. Sieve Methods, Exponential Sums, and Their Applications in Number Theory (Cardiff, 1995), 1997, Cambridge. Cambridge Univ. Press. 87100. 237

[3] CookR. The value of additive forms at prime arguments. J. Théor. Nombres Bordeaux, 2001, 13(1): 77-91.

[4] CookR, FoxA. The values of ternary quadratic forms at prime arguments. Mathematika, 2001, 48(1–2): 137-149.

[5] CookR, HarmanG. The values of additive forms at prime arguments. Rocky Mountain J. Math., 2006, 36(4): 1153-1164.

[6] DavenportH, HeilbronnH. On indefinite quadratic forms in five variables. J. London Math. Soc., 1946, 21: 185-193.

[7] GambiniA. Diophantine approximation with one prime, two squares of primes and one k-th power of a prime. Open Math., 2021, 19(1): 373-387.

[8] GambiniA, LanguascoA, ZaccagniniA. A Diophantine approximation problem with two primes and one k-power of a prime. J. Number Theory, 2018, 188: 210-228.

[9] GaoG, LiuZ. Results of Diophantine approximation by unlike powers of primes. Front. Math. China, 2018, 13(4): 797-808.

[10] Ge W., Li W., One Diophantine inequality with unlike powers of prime variables. J. Inequal. Appl., 2016, 2016: Paper No. 33, 8 pp.

[11] HarmanG. Diophantine approximation by prime numbers. J. London Math. Soc. (2), 1991, 44(2): 218-226.

[12] HarmanG. The values of ternary quadratic forms at prime arguments. Mathematika, 2004, 51(1–2): 83-96.

[13] HarmanG, KumchevAV. On sums of squares of primes. Math. Proc. Cambridge Philos. Soc., 2006, 140(1): 1-13.

[14] LanguascoA, ZaccagniniA. A Diophantine problem with a prime and three squares of primes. J. Number Theory, 2012, 132(12): 3016-3028.

[15] LanguascoA, ZaccagniniA. On a ternary Diophantine problem with mixed powers of primes. Acta Arith., 2013, 159(4): 345-362.

[16] LanguascoA, ZaccagniniA. A Diophantine problem with prime variables. Highly Composite: Papers in Number Theory, 2016, Mysore. Ramanujan Math. Soc.. 15716823

[17] LiW, WangT. Diophantine approximation with one prime and three squares of primes. Ramanujan J., 2011, 25(3): 343-357.

[18] LiuZ, SunH. Diophantine approximation with one prime and three squares of primes. Ramanujan J., 2013, 30(3): 327-340.

[19] MatomäkiK. Diophantine approximation by primes. Glasg. Math. J., 2010, 52(1): 87-106.

[20] MuQ. Diophantine approximation with four squares and one kth power of primes. Ramanujan J., 2016, 39(3): 481-496.

[21] MuQ, ZhuM, LiP. A Diophantine inequality with four squares and one kth power of primes. Czechoslovak Math. J., 2019, 69(2): 353-363.

[22] RiegerG. Über die Summe aus einem Quadrat und einem Primzahlquadrat. J. Reine Angew. Math., 1968, 231: 89-100

[23] VaughanR. Diophantine approximation by prime numbers, I. Proc. London Math. Soc. (3), 1974, 28: 373-384.

[24] VaughanR. Diophantine approximation by prime numbers, II. Proc. London Math. Soc. (3), 1974, 28: 385-401.

[25] WangY, YaoW. Diophantine approximation with one prime and three squares of primes. J. Number Theory, 2017, 180: 234-250.

[26] WatsonG. On indefinite quadratic forms in five variables. Proc. London Math. Soc. (3), 1953, 3: 170-181.

RIGHTS & PERMISSIONS

Peking University