Number Theory: An Introduction to Mathematics

1 Uniform Distribution 455

is a polynomial of degreer−1 with leading coefficientrmαr.Ifαris irrational, then
rmαris also irrational and hence, by the induction hypothesis, the sequence(gm(n))
is uniformly distributed mod 1. Consequently, by Proposition 6, the sequence(f(n))
is also uniformly distributed mod 1.
Suppose next that the leading coefficientαris rational, and letαs( 1 ≤s<r)be
the coefficient nearest to it which isirrational. Then the coefficients oftr−^1 ,...,tsof
the polynomialgm(t)are rational, but the coefficient ofts−^1 is irrational. Ifs>1,
it follows again from the induction hypothesis and Proposition 6 that the sequence
(f(n))is uniformly distributed mod 1.
Suppose finally thats=1 and put

F(t)=αrtr+αr− 1 tr−^1 +···+α 2 t^2.

Ifq>0 is a common denominator for the rational numbersα 2 ,...,αrthen, for any

integerh=0 and any nonnegative integersj,k,

e(hF(jq+k))=e(hF(k)).

WriteN=q+k,where=N/qand 0≤k<q.Sincef(t)=F(t)+α 1 t+α 0 ,

we obtain

N−^1

N∑− 1

n= 0

e(hf(n))=N−^1

q∑− 1

k= 0

∑− 1

j= 0

e(hf(jq+k))+N−^1

∑N

n=q

e(hf(n))

=N−^1 N/q

q∑− 1

k= 0

e(hF(k))

∑− 1

j= 0

−^1 e(h(jqα 1 +kα 1 +α 0 ))

+N−^1

∑N

n=q

e(hf(n)).

The last term tends to zero asN→∞, since the sum contains at mostqterms, each

of absolute value 1. By Theorem 2, each of theqinner sums in the first term also tends

to zero asN→∞, because the result holds forr=1. Hence, by Theorem 2 again,

the sequence(f(n))is uniformly distributed mod 1. 

An interesting extension of Proposition 6 was derived by Korobov and Postnikov

(1952):

Proposition 8If, for every positive integer m, the sequence(ξn+m−ξn)is uniformly

distributed mod 1 then, for all integers q> 0 and r ≥ 0 , the sequence(ξqn+r)is

uniformly distributed mod 1.

Proof We may supposeq>1, since the assertion follows at once from Proposition 6

ifq=1. By Theorem 2 it is enough to show that, for every integerm=0,

S:=N−^1

∑N

n= 1

e(mξqn+r)→0asN→∞.