In this paper we study the distribution of lattice points on arcs of circles centered at the origin. We show that on such a circle of radius
R
R
, an arc whose length is smaller than
2
R
1
/
2
−
1
(
4
[
m
/
2
]
+
2
)
\sqrt 2 {R^{1/2 - 1(4[m/2] + 2)}}
contains, at most,
m
m
lattice points. We use the same method to obtain sharp
L
4
{L^4}
-estimates for uncompleted, Gaussian sums