1.4. Equidistribution in finite fields 69

Proposition 1.4.8.

R[D]

is equidistributed in

Σ[D],

thus

|{(x, h1,...,hD) ∈ V

d+1

:

R[D](x,

h1,...,hD) = r}|

=

1

|Σ[D]|

+ o(1) |V

|d+1

for all r ∈

Σ[D].

Furthermore, we have the refined bound

|{(h1,...,hD) ∈ V

d

:

R[D](x,

h1,...,hD) = r}|

=

pm

|Σ[D]|

+ o(1) |V

|d

for all r ∈

Σ[D]

and all x ∈ Sr0 .

Proof. It suﬃces to prove the second claim. Fix x and r = (rω)ω∈{0,1}D .

From the definition of

Σ[D],

we see that r is uniquely determined by the

component r0 and rd := (rω)ω∈{0,1}D:0 |ω|d. It will thus suﬃce to show

that

|{(x, h1,...,hD) ∈ V

d

:

Rd](x, [D

h1,...,hD) = rd}|

=

pm

|Σ[D]|

+ o(1) |V

|d

for all rd ∈

(Fm){ω∈{0,1}D:0 |ω|d},

where

Rd](x, [D

h1,...,hD)

:= (R(x + ω1h1 + · · · + ωDhD))ω∈{0,1}D:0

|ω|d

.

By Fourier analysis, it suﬃces to show that

Eh1,...,hD∈V e ξ ·

Rd](x, [D

h1,...,hD) = o(1)

for any non-zero ξ ∈

(Fm){ω∈{0,1}D:0

|ω|d}.

In other words, we need to

show that

(1.39)

Eh1,...,hD∈V e

⎛

⎝

ω∈{0,1}D:|ω|d

ξω · R(x + ω1h1 + · · · +

ωDhD)⎠

⎞

= o(1)

whenever the ξω ∈

Fm

for ω ∈ {0,

1}D,

0 |ω| d are not all zero.

Let ω0 be such that ξω0 = 0, and such that |ω| is as large as possible; let

us write d := |ω0|, so that 0 ≤ d d. Without loss of generality, we may

take ω0 = (1,..., 1, 0,..., 0). Suppose (1.39) failed, then by the pigeonhole

principle one can find hd +1,...,hD such that

|Eh1,...,hd

∈V

e(

ω∈{0,1}D:|ω|d

ξω · R(x + ω1h1 + · · · + ωDhD))| 1.