| 牟全武,吕晓东.素变量混合幂丢番图逼近[J].数学年刊A辑,2015,36(3):303~312 |
| 素变量混合幂丢番图逼近 |
| Diophantine Approximation with Prime Variablesand Mixed Powers |
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| DOI： |
| 中文关键词: 丢番图不等式, Davenport-Heilbronn\,方法, 素数 |
| 英文关键词:Diophantine inequalities, Davenport-Heilbronn method, Prime |
| 基金项目:本文受到国家自然科学基金 (No.11201107, No.11271283)和陕西省自然科学基础研究计划 (No.2014JM2-1010)的资助 |
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| 中文摘要: |
| 设~$\lambda_1, \lambda_2, \lambda_3, \lambda_4$是正实数, $\frac{\lambda_1}{\lambda_2}$是无理数和代数数,
$\mathcal {V}$是具有良好间隔的序列, $\delta>0$. 证明了: 对于任意的$\varepsilon>0$及$v\in \mathcal {V},\ v\leq X$,
使得$|\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^3+\lambda_4p_4^3-v| |
| 英文摘要: |
| Let λ1, λ2, λ3, λ4 be positive real numbers, and let λ1
λ2
be irrational and algebraic.
Let V be a well-spaced sequence, δ > 0. It is proved that for any given ε > 0, the number
of v satisfying v ∈ V and v 6 X for which |λ1p21
+ λ2p22
+ λ3p33
+ λ4p34
? v| < v?δ has no
solution in primes p1, p2, p3, p4 does not exceed O(X
67
72+2δ+ε). This gives an improvement
of the earlier result. |
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