Some new theorems in metric diophantine approximation are obtained by dyadic methods. We show for example that if
m
1
,
m
2
,
…
{m_1},{m_2}, \ldots
, are distinct integers with
m
n
=
O
(
n
p
)
{m_n} = O({n^p})
then
Σ
n
⩽
N
e
(
m
n
x
)
=
O
(
N
1
−
q
)
{\Sigma _{n \leqslant N}}e({m_n}x) = O({N^{1 - q}})
except for a set of x of Hausdorff dimension at most
(
p
+
4
q
−
1
)
/
(
p
+
2
q
)
(p + 4q - 1)/(p + 2q)
; and that for any sequence of intervals
I
1
,
I
2
,
…
{I_1},{I_2}, \ldots
in [0, 1) the number of solutions of
{
x
n
}
∈
I
n
(
n
⩽
N
)
\{ {x^n}\} \in {I_n}\;(n \leqslant N)
is a.e. asymptotic to
Σ
n
⩽
N
|
I
n
|
(
x
>
1
)
{\Sigma _{n \leqslant N}}|{I_n}|(x > 1)
.