1. Heilbronn' s Theore m
Dirichlet's classica l Theore m say s tha t for real a and for N 1, there is a
natural n N and an integer m with
\an- m\ AT 1 .
Write ||s| | fo r th e distanc e fro m a rea l numbe r f t o th e neares t integer . Dirichlet' s
Theorem say s tha t for real a and for N 1, there is a natural n with n N and
(1-1) | | a
B
| | N - 1.
Considerable difficultie s aris e i f th e linea r functio n fin) = CLn i s replace d b y othe r
functions o f n. Heilbron n [1948], improvin g a resul t o f Vinogrado v [1927], prove d
the followin g
Theorem 1A. Suppose 0 is given, and N Cju) . Then for any real a , there
is a natural n N with
(1.2) \\an 2\\N-1/2+.
In Dirichlet' s Theorem , th e inequalit y (1.1) i s know n t o b e bes t possible . I n
Heilbronn's Theore m i t i s conceivabl e tha t th e exponen t -Vi + i n (1.2) ma y b e re -
placed b y - 1 + e .
Suppose w e hav e a hypothetica l Heilbronn' s Theore m wit h (1.2) replace d b y
(1.3) ||an
2||A/V)~\
where f\N) i s increasing . Fo r give n n, (1.3) define s a subse t o f th e uni t interva l
0 . a 1 o f measur e 2/(A/) ~ , an d fo r 1 nN, th e unio n o f thes e subset s ha s
measure .v . 2N/(N)-1 . I f (1.3) i s tru e fo r ever y a , the n 1 2Nf(N)' X an d f(N) 2N.
So th e righ t han d sid e i n (1.2) ma y no t b e replace d b y a quantit y les s tha n (2N)~ . I n
fact, w e ca n sa y more :
Let N ~ p 1 wher e f i s a n od d prime , an d a = a/p wher e a is a quadrati c non
residue modul o p. Th e relatio n (1.3) implie s tha t an ^ b (mo d p) wit h \b\ pf(N)~ .
Thus ther e i s a quadrati c nonresidu e b modul o p wit h
References a t th e end ar e listed b y the nam e of th e author , the n b y the yea r o f publica -
tion.
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