I consider this set $S_{0}=\{ n \in \mathbb{N}: n\sin^2(n) < 1 \}$. And I have some questions.
See the elements of $S_{0}$=
$\{ 1,3,6,19,22,25,44,47,66,69,88,110,132,154,157,176,179,$ (common difference 22)
$201,223,245,267,289,311,333,355,377,399,421,443,465,$ (common difference 22)
$622,644,666, 688,710,732,754,776, $ (common difference 22)
$999,1021, 1043, 1065, 1087,1109, 1131 $ (common difference 22)
$1354, 1376, 1398, 1420, 1442, 1464 $ (common difference 22)
$1731,1753, 1775,1797, 1819, $ (common difference 22)
$2086,2108,2130,2152,2174 $ (common difference 22)
$2441,2463,2485,2507,2529 $ (common difference 22)
$\ldots$
$19170,\ldots,19170 +355t,\ldots,51475$
(no elements)
$53228,\ldots,53228 +355t,\ldots,135943$ (size: 82715)
(no elements) (size: 45063)
$181006,\ldots,181006 +355t,\ldots,232836$ (size: 51830)
(no elements) (size: 58198)
$291034,\ldots,291034 +355t,\ldots,332924$ (size: 41890)
(no elements) (size: 65298)
$398222,\ldots,398222 +355t,\ldots,434432$ (size: 36210)
(no elements) (size: 70268)
$504700,\ldots,504700 +355t,\ldots,537005$ (size: 32305)
(no elements) (size: 73463)
$610468,\ldots,610468 +355t,\ldots,639933$ (size: 29465)
(no elements) (size: 75948)
$715881,\ldots,715881 +355t,\ldots,742861$ (size: 26980)
(no elements) (size: 78078)
$820939,\ldots,820939 +355t,\ldots,846499$ (size: 25560)
(no elements) (size: 79498)
$925997,\ldots,925997 +355t,\ldots,949782$ (size: 23785)
$\ldots.\}$
Note that the common difference of the some elements of $S_{0}$ are 22 (for the first elements) and 355 (for after elements), but I don't know the reason why.
Note that the size of the set of elements that have common difference 355 is decreasing, and the distance of one to another is becoming large (it's the set "no elements" that is becoming large). So, I conjectured that there is $n_{\delta} \notin S_{0}$ such that for all $n > n_{\delta}$, if $n \in S_{0}$, then $n + 355 \notin S_{0}$. And if $n_{1},n_{2} \in S_{0}$, such that $n_{1},n_{2} > n_{\delta}$, then $n_{1}<<n_{2}$.
Considering the element of $S_{0}$ that begin subsequence of common difference 355, we have: $A =\{53228,181006,291034,398222,504700,610468,715881,820939,925997,\ldots\}$
Apply $|\sin(n)|$,
$\{0,004329758;0,002329227;0,001835904;0,001583734;0,001391853;0,001260261;0,001158813;0,001087509;0,001016206,\ldots\}$
If there is $n_{\delta}$, so $A$ is finite.
Is $sin(n) \to 0$ for $n \in S_{0}$ such that $n > n_{\delta}$ ?