Affiliation:
1. Dipartimento di Matematica ‘Guido Castelnuovo’, P.le Aldo Moro 5, Sapienza Università di Roma , Rome , Italy
Abstract
Abstract
Let
1
<
k
<
14
/
5
1\lt k\lt 14\hspace{-0.08em}\text{/}\hspace{-0.08em}5
,
λ
1
,
λ
2
,
λ
3
{\lambda }_{1},{\lambda }_{2},{\lambda }_{3}
and
λ
4
{\lambda }_{4}
be non-zero real numbers, not all of the same sign such that
λ
1
/
λ
2
{\lambda }_{1}\hspace{-0.08em}\text{/}\hspace{-0.08em}{\lambda }_{2}
is irrational and let
ω
\omega
be a real number. We prove that the inequality
∣
λ
1
p
1
+
λ
2
p
2
2
+
λ
3
p
3
2
+
λ
4
p
4
k
−
ω
∣
≤
(
max
(
p
1
,
p
2
2
,
p
3
2
,
p
4
k
)
)
−
ψ
(
k
)
+
ε
| {\lambda }_{1}{p}_{1}+{\lambda }_{2}{p}_{2}^{2}+{\lambda }_{3}{p}_{3}^{2}+{\lambda }_{4}{p}_{4}^{k}-\omega | \le {\left(\max \left({p}_{1},{p}_{2}^{2},{p}_{3}^{2},{p}_{4}^{k}))}^{-\psi \left(k)+\varepsilon }
has infinitely many solutions in prime variables
p
1
,
p
2
,
p
3
,
p
4
{p}_{1},{p}_{2},{p}_{3},{p}_{4}
for any
ε
>
0
\varepsilon \gt 0
, where
ψ
(
k
)
=
min
1
14
,
14
−
5
k
28
k
\psi \left(k)=\min \left(\frac{1}{14},\frac{14-5k}{28k}\right)
.