The well-known Sidon-Telyakovskii integrability condition is considerably lightened as follows:
\[
1
n
∑
k
=
1
n
|
Δ
c
(
k
)
|
p
A
k
p
=
O
(
1
)
,
n
→
∞
,
\frac {1}{n}\sum \limits _{k = 1}^n {\frac {{|\Delta c(k){|^p}}}{{A_k^p}} = O(1),\quad n \to \infty } ,
\]
where
{
c
(
n
)
}
\{ c(n)\}
is a certain null-sequence and
1
>
p
≤
2
1 > p \leq 2
. It is proved that
∑
n
=
1
∞
n
p
−
1
|
Δ
c
(
n
)
|
p
ρ
p
(
n
)
>
∞
\sum \nolimits _{n = 1}^\infty {{n^{p - 1}}|\Delta c(n){|^p}{\rho ^p}(n) > \infty }
is also a sufficient integrability condition provided
∑
n
=
1
∞
(
1
/
n
ρ
(
n
)
)
>
∞
\sum \nolimits _{n = 1}^\infty {(1/n\rho (n)) > \infty }
, where
{
ρ
(
n
)
}
\{ \rho (n)\}
is an increasing sequence of positive numbers.