SMALL FRACTIONA L PART S O F POLYNOMIAL S
That on e canno t possibl y g o muc h furthe r i s see n fro m th e followin g f 'metrical*'
result, whic h i s a specia l cas e o f a theore m o f Schmid t L1964J.
Let xjj(n) 0 be decreasing to 0 . The inequality ||CXT 2 | | if/{n) has only finitely
many solutions for almost every CL if the sum H _
1
l/f(w) is convergent, and it has
infinitely many solutions for almost every CL if the sum is divergent.
In particular , takin g if/(n) = 1/(72 log n log log n), w e se e tha t (1.7) i s tru e fo r alm
every a . Bu t takin g ift(n) = l/(n(\ogn) ) , w e se e tha t
lim inf nilogn) \\n CL\\ =0 0
7 2 - * OO
for almos t ever y CL .
It i s wel l know n tha t
lim in f 72||T2CL| | = 0
n—»oo
precisely i f th e partia l denominator s i n th e continue d fractio n expansio n o f CL are un
bounded. Bu t n o analogou s resul t i s know n concernin g (1.6).
After givin g a proo f o f Heilbronn' s Theore m i n §§2—6 , w e shal l tur n i n § 7 t o th
more genera l question s whic h aris e whe n dn i s replace d b y a mor e genera l polynom
f(n). I n §13 w e wil l tak e u p simultaneou s approximatio n o f quadrati c polynomials , an
in §2 0 certai n quadrati c polynomial s i n severa l variables .
2. Th e Heilbron n Alternativ e Lemm a
A rea l numbe r q i s uniquel y writte n a s
£=[£] +
{£!
where [£] , th e integer part o f £, i s a n integer , an d wher e i f } , th e fractiona l par t o f
f, satisfie s 0 \q } 1. Le t 12 b e th e uni t interval , consistin g o f number s £ i n
0 1 .
Lemma 2A . Suppose e 0, N Cj(e) 5 and N~ l/2+€ I 1. Let CL be arbitrary.
Then either every subinterval 3 of 12 of length I contains an element \cLn 1 with
1 n N. Or there is a natural
qI"2N€ with \\aq\\ l~ lN€~2.
Heilbronn's Theore m 1A follows : Se t / = N~ l/2+€. I n th e firs t alternative , ther
a natura l n N wit h 0 \an 2\ 1, whenc e wit h ||^w 2 || N~ 1/2+*. I n th e secon d al
ternative w e hav e q N ~ N an d
4 Ou r proo f o f Heilbronn' s Theore m follow s th e argumen t o f Heilbron n a s presente d b y
Davenport [1967], bu t a n Alternativ e Lemm a wa s no t explicitl y formulate d b y thes e authors . W
call th e lemm a th e Heilbron n A . L . t o distinguis h i t fro m othe r alternativ e lemma s comin g u p
in th e sequel .
The numberin g o f constant s i s starte d ne w i n eac h section .
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