Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Fundamental Domains
for Rational Function Based Digit Systemsof Formal Laurent Series over Finite Fields
Alfonso David C. Rodriguez(joint work with M. J. C. Loquias)
Institute of Mathematics
University of the Philippines Diliman

Numeration 2018
May 23, 2018

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Outline1

Number systems and tilings over Laurent series
Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Fundamental Domains

2

Rational function based digit systems over finite fields
P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))3Connection
Connecting P/Q-digit systems to tilings over Laurent series
Fundamental Domains for P/Q-digit systems

4

Recommendations for future work

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Fundamental Domains

Outline1

Number systems and tilings over Laurent series
Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Fundamental Domains

2

Rational function based digit systems over finite fields
P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))3Connection
Connecting P/Q-digit systems to tilings over Laurent series
Fundamental Domains for P/Q-digit systems

4

Recommendations for future work

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Fundamental Domains

Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Given:
F a fieldf ∈ F[x, y] monic in both x and ys ∈ S ∶= F((x−1 , y−1 ))/f F((x−1 , y−1 ))x-digit representation of s : h = ∑ki=−∞ di xi ∈ F((x−1 , y−1 )) suchthat s = h + f F((x−1 , y−1 )), k ∈ Z and di ∈ F[y] withdegy (di ) < degy (f )Theorem (Beck-Brunotte-Scheicher-Thuswaldner, 2009)
Let f ∈ F[x, y] be monic in both x and y. Then every element of S
admits a unique x-digit representation.

.

x-digit string of s : ⟪s⟫x = dk dk−1 ⋯d0 d−1 ⋯Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Fundamental Domains

Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Given:
F a fieldf ∈ F[x, y] monic in both x and ys ∈ S ∶= F((x−1 , y−1 ))/f F((x−1 , y−1 ))x-digit representation of s : h = ∑ki=−∞ di xi ∈ F((x−1 , y−1 )) suchthat s = h + f F((x−1 , y−1 )), k ∈ Z and di ∈ F[y] withdegy (di ) < degy (f )Theorem (Beck-Brunotte-Scheicher-Thuswaldner, 2009)
Let f ∈ F[x, y] be monic in both x and y. Then every element of S
admits a unique x-digit representation.

.

x-digit string of s : ⟪s⟫x = dk dk−1 ⋯d0 d−1 ⋯Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Fundamental Domains

Example of x-digit representation
ExampleGiven:
F = F5f = y3 + (2x + 1)y + 3x + x2 ∈ F[x, y]s = y5 + 2x4 y4 + f F((x−1 , y−1 ))We get:y5 + 2x4 y4 ≡ (3y)x6 + (y2 + 4y)x5 + (3y2 )x4 + (2)x3+ (4y2 + 4y + 2)x2 + (2y2 + 4y + 3)x + y (mod f ).

⟪s⟫x = 3y, y2 + 4y, 3y2 , 2, 4y2 + 4y + 2, 2y2 + 4y + 3, y 0

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Fundamental Domains

Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Given: s ∈ S with x-digit representation h ∈ F((x−1 , y−1 ))x-height of s : hgtx (s) = degx (h)⎧⎪⎪#F if F is finite
x-norm : ∣s∣x ∶= q
where q = ⎨⎪if F is infinite⎪⎩e
S is a complete topological F((x−1 ))-vector space wrt ∣ ⋅ ∣x
hgtx (s)x-fundamental domain of S : Fx ∶= {s ∈ S ∣ ∣s∣x < 1}

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Fundamental Domains

Mutual composition of fundamental domains
Proposition (Beck-Brunotte-Scheicher-Thuswaldner, 2009)
Set m = degx (f ), n = degy (f ), ρ = (m − 1)(n − 1), and⎧⎫RRR
m−1 n−1⎪⎪⎪⎪Vx ∶= ⎨s ∈ Fx RRRRR s = ∑ ∑ aij xi yj + f F((x−1 , y−1 )) for some aij ∈ F⎬ .
⎪
⎪
RRR
⎪
⎪
i=0 j=−ρ⎩⎭Then Vx is an F-vector space andFx = ⋃ (s + y−ρ Fy ).
s∈Vx

In particular, Fx is a clopen, bounded, and compact subset of the
topological F((y−1 ))-vector space S.
Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Fundamental Domains

Tilings induced by fundamental domains

Theorem (Beck-Brunotte-Scheicher-Thuswaldner, 2009)
The collection {r + Fx ∣ r ∈ F[x, y]/f F[x, y]} forms a tiling of theF((y−1 ))-space S in the sense thatS=⋃

(r + Fx )r∈F[x,y]/f F[x,y]and (r + Fx ) ∩ (r′ + Fx ) = ∅ for any distinct r, r′ ∈ F[x, y]/f F[x, y].

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Fundamental Domains

Visualization of fundamental domains when F is finite

Suppose F = Fq is finite of order q. Let u ∶ F Ð→ {0, . . . , q − 1} be an
enumeration of F. Consider the map µ × ⋯ × µ ∶ F((y−1 ))m Ð→ Rm´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶m times

where µ is defined byµ ∶ F((y−1 )) Ð→ R
l

i

l

i

∑ ai y z→ ∑ u(ai )q .
i=−∞

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)i=−∞

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Fundamental Domains

Visualization of fundamental domains when F is finite
Examplef = x3 + (y + 1)x2 + x + y2 + y ∈ F2 [x, y]Ô⇒ n = 2, Vy has an F2 -basis {x−2 , x−1 , 1 + x−2 y, x + x−1 y} (mod f )Ô⇒ ⟨(x−2 , 0), (x−1 , 0), (1, x−2 ), (x, x−1 )⟩F ⊂ F((x−1 ))2Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))Outline1

Number systems and tilings over Laurent series
Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Fundamental Domains

2

Rational function based digit systems over finite fields
P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))3Connection
Connecting P/Q-digit systems to tilings over Laurent series
Fundamental Domains for P/Q-digit systems

4

Recommendations for future work

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))P/Q-digit systems in F[x]Given:
F = Fq a finite field of order q
P, Q ∈ F[x] coprime with deg P > deg Q > 0
w ∈ F[x]P/Q-polynomial digit expansion of w :ksi P i( )i=0 Q Qw=∑

where k ≥ 0 and si ∈ D ∶= {s ∈ F[x] ∣ deg s < deg P}.
Theorem (Loquias-Mkaouar-Scheicher-Thuswaldner, 2017)
Let P, Q ∈ F[x] be coprime with deg P > deg Q > 0. Then every
element of F[x] admits a unique P/Q-polynomial digit expansion.
Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))P/Q-digit systems in F[x]Given:
F = Fq a finite field of order q
P, Q ∈ F[x] coprime with deg P > deg Q > 0
w ∈ F[x]P/Q-polynomial digit expansion of w :ksi P i( )i=0 Q Qw=∑

where k ≥ 0 and si ∈ D ∶= {s ∈ F[x] ∣ deg s < deg P}.
Theorem (Loquias-Mkaouar-Scheicher-Thuswaldner, 2017)
Let P, Q ∈ F[x] be coprime with deg P > deg Q > 0. Then every
element of F[x] admits a unique P/Q-polynomial digit expansion.
Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))ExamplesExampleGiven F = F2 , P = x2 + 1, Q = xx2 + x =x3 + 1 =x+1
x P 2 x+1 P
( ) +
( )+
Q Q
Q
Q
Q

x P 3 1 P 2 0 P
x+1( ) + ( ) + ( )+
Q Q
Q Q
Q Q
Q

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))P/Q-digit systems in F[x]expansion tree T(P/Q) : edge-labeled infinite directed graph(F[x], E) where (v, w) ∈ E with label s ∈ D if and only if
Pv + s.
w=
Q
Theorem (Loquias-Mkaouar-Scheicher-Thuswaldner, 2017)
T(P/Q) is qdeg P−deg Q -regular with a loop at the root 0.
ω-language WP/Q of T(P/Q) : set of all right-infinite strings
formed from the labels of infinite paths starting from the root of
T(P/Q)
Theorem (Loquias-Mkaouar-Scheicher-Thuswaldner, 2017)
The only eventually periodic string in WP/Q is 0ω = 00⋯.
Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))P/Q-digit systems in F[x]expansion tree T(P/Q) : edge-labeled infinite directed graph(F[x], E) where (v, w) ∈ E with label s ∈ D if and only if
Pv + s.
w=
Q
Theorem (Loquias-Mkaouar-Scheicher-Thuswaldner, 2017)
T(P/Q) is qdeg P−deg Q -regular with a loop at the root 0.
ω-language WP/Q of T(P/Q) : set of all right-infinite strings
formed from the labels of infinite paths starting from the root of
T(P/Q)
Theorem (Loquias-Mkaouar-Scheicher-Thuswaldner, 2017)
The only eventually periodic string in WP/Q is 0ω = 00⋯.
Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))P/Q-digit systems in F[x]expansion tree T(P/Q) : edge-labeled infinite directed graph(F[x], E) where (v, w) ∈ E with label s ∈ D if and only if
Pv + s.
w=
Q
Theorem (Loquias-Mkaouar-Scheicher-Thuswaldner, 2017)
T(P/Q) is qdeg P−deg Q -regular with a loop at the root 0.
ω-language WP/Q of T(P/Q) : set of all right-infinite strings
formed from the labels of infinite paths starting from the root of
T(P/Q)
Theorem (Loquias-Mkaouar-Scheicher-Thuswaldner, 2017)
The only eventually periodic string in WP/Q is 0ω = 00⋯.
Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))P/Q-digit systems in F[x]expansion tree T(P/Q) : edge-labeled infinite directed graph(F[x], E) where (v, w) ∈ E with label s ∈ D if and only if
Pv + s.
w=
Q
Theorem (Loquias-Mkaouar-Scheicher-Thuswaldner, 2017)
T(P/Q) is qdeg P−deg Q -regular with a loop at the root 0.
ω-language WP/Q of T(P/Q) : set of all right-infinite strings
formed from the labels of infinite paths starting from the root of
T(P/Q)
Theorem (Loquias-Mkaouar-Scheicher-Thuswaldner, 2017)
The only eventually periodic string in WP/Q is 0ω = 00⋯.
Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))The expansion tree T(P/Q) for P = x2 + 1, Q = x ∈ F2 [x]xx3 + x2 + x

0x3 + x2 + x + 1

2

x+1 x +x
x+1
1

x+1
0

0

x

1

1

x3 + x2 + 1⋯xx3 + x + 1⋯

0x3 + x⋯x+1

x3 + 1

⋯

1

x3

⋯

2

x
0

x2 + 1x

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

x3 + x2

x +x+1

x

⋯

x+1
2

1

⋯

⋯

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))P/Q-digit systems in F((x−1 )Given:
F = Fq a finite field of order q
P, Q ∈ F[x] coprime with deg P > deg Q > 0α ∈ F((x−1 ))P/Q-series digit expansion of α :ksi P i( )i=−∞ Q Q

α= ∑where k ∈ Z and sk sk−1 ⋯ ∈ WP/Q
Theorem (Loquias-Mkaouar-Scheicher-Thuswaldner, 2017)
Let P, Q ∈ F[x] be coprime with deg P > deg Q > 0. Then every
element of F((x−1 )) admits a unique P/Q-series digit expansion.

.

P/Q-series digit string of α : ⟪α⟫P/Q ∶= sk sk−1 ⋯s0 s−1 ⋯Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))P/Q-digit systems in F((x−1 )Given:
F = Fq a finite field of order q
P, Q ∈ F[x] coprime with deg P > deg Q > 0α ∈ F((x−1 ))P/Q-series digit expansion of α :ksi P i( )i=−∞ Q Q

α= ∑where k ∈ Z and sk sk−1 ⋯ ∈ WP/Q
Theorem (Loquias-Mkaouar-Scheicher-Thuswaldner, 2017)
Let P, Q ∈ F[x] be coprime with deg P > deg Q > 0. Then every
element of F((x−1 )) admits a unique P/Q-series digit expansion.

.

P/Q-series digit string of α : ⟪α⟫P/Q ∶= sk sk−1 ⋯s0 s−1 ⋯Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))ExampleExampleGiven F = F2 , P = x2 = 1, Q = x.
⟪x +1⟫ = x, 1, 0, x+1.1, x, 0, x+1, x, 1, 0, x+1, 0, 0, 0, 0, x, 1, 0, x+1, . . .

⟪x2 +x⟫ = x, x+1, x+1 0, 1, x+1, x+1, 0, x, x+1, x+1, 0, 0, 0, 0, 0, x, . . .
3Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Connecting P/Q-digit systems to tilings over Laurent series
Fundamental Domains for P/Q-digit systems

Outline1

Number systems and tilings over Laurent series
Digit systems in F((x−1 , y−1 ))/f F((x−1 , y−1 ))Fundamental Domains

2

Rational function based digit systems over finite fields
P/Q-digit systems in F[x]P/Q-digit systems in F((x−1 ))3Connection
Connecting P/Q-digit systems to tilings over Laurent series
Fundamental Domains for P/Q-digit systems

4

Recommendations for future work

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Connecting P/Q-digit systems to tilings over Laurent series
Fundamental Domains for P/Q-digit systems

Connecting P/Q-digit systems to tilings over Laurent series
f = Qy − P (non-monic in y if deg Q > 0)s ∈ S ∶= F((x−1 , y−1 ))/f F((x−1 , y−1 ))x-digit representation of s : α ∈ F((x−1 )) such thats = α + f F((x−1 , y−1 ))y-digit representation of s :
k

si i
y
i=−∞ Qdy (α) ∶= ∑
where

k

si P i( )i=−∞ Q Q

α= ∑is the P/Q-series digit expansion of α.
Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Connecting P/Q-digit systems to tilings over Laurent series
Fundamental Domains for P/Q-digit systems

Connecting P/Q-digit systems to tilings over Laurent series
Fix m = deg P, n = deg Q.
0m−1

Consider the vector space Vy = spanF((y−1 )) { xQ , . . . , x Q }Let E ∶= {dy (α) ∣ α ∈ F((x−1 ))}Proposition (L.-R., 20xx)
E is a proper subspace of Vy with dimension m − n over F((y−1 )).
Moreover, a basis for E is any set {dy (α1 ), . . . , dy (αm−n )} where
α1 , . . . , αm−n ∈ F((x−1 )) such that the left-most digits of the digit
strings ⟪αi ⟫P/Q are linearly independent over F.

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Connecting P/Q-digit systems to tilings over Laurent series
Fundamental Domains for P/Q-digit systems

ExamplesExampleGiven P = x2 + 1, Q = x ∈ F2 [x]B = {dy (1 + x12 )}.

⟪1 + x12 ⟫P/Q = x 1, 0, 1, 0, 0, 0, 1, s−8 , s−9 , . . . where⎧⎪⎪1 if i = 2n − 1 for some n ∈ N
s−i ∶= ⎨⎪⎪⎩0 otherwise∞n∴ E = spanF((y−1 )) {( ∑ y1−2 ) (n=1

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)x1
x0) + (1) ( )}Q
Q

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Connecting P/Q-digit systems to tilings over Laurent series
Fundamental Domains for P/Q-digit systems

ExamplesExampleGiven P = x3 + 1, Q = x ∈ F2 [x]B = {dy (α1 ), dy (α2 )}.
⟪α ⟫= x .0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, . . .
⟪α1 ⟫P/Q = x 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, . . .
1 P/Q

2

∴ dy (α1 ) = (y−1 + y−4 + y−7 + y−13 + ⋯) (∴ dy (α2 ) = (y−2 + y−8 + y−14 + ⋯) (Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

x0
x1
x2
) + (1) ( ) + (0) ( )
Q
Q
Q

x0
x1
x2
) + (0) ( ) + (1) ( )Q
Q
Q
Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Connecting P/Q-digit systems to tilings over Laurent series
Fundamental Domains for P/Q-digit systems

Fundamental Domains for P/Q-digit systems
y-height of s : hgty (s) = degy (dy (α))⎧⎪⎪#F if F is finite
y-norm : ∣s∣y ∶= qhgty (s) where q = ⎨⎪if F is infinite⎪⎩e
S is a complete topological F((y−1 ))-vector space wrt ∣ ⋅ ∣yS ≅ E as F((y−1 ))-vector spaces

y-fundamental domain of S : Fy ∶= {s ∈ S ∣ ∣s∣y < 1}
Proposition (L.-R., 20xx)
Fx = Fy

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Connecting P/Q-digit systems to tilings over Laurent series
Fundamental Domains for P/Q-digit systems

Fundamental Domains for P/Q-digit systems
y-height of s : hgty (s) = degy (dy (α))⎧⎪⎪#F if F is finite
y-norm : ∣s∣y ∶= qhgty (s) where q = ⎨⎪if F is infinite⎪⎩e
S is a complete topological F((y−1 ))-vector space wrt ∣ ⋅ ∣yS ≅ E as F((y−1 ))-vector spaces

y-fundamental domain of S : Fy ∶= {s ∈ S ∣ ∣s∣y < 1}
Proposition (L.-R., 20xx)
Fx = Fy

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Connecting P/Q-digit systems to tilings over Laurent series
Fundamental Domains for P/Q-digit systems

ExampleExampleGiven P = x3 + 1, Q = x ∈ F2 [x]Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

Recommendations for future work

Find suitable digit systems for F((x−1 , y−1 ))/f F((x−1 , y−1 ))where f is non-monic in at least one indeterminate andnon-linear in both indeterminates
Find suitable digit systems for F((x−1 , y−1 ))/f F((x−1 , y−1 ))where f is non-monic in both indeterminates

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields

Number systems and tilings over Laurent series
Rational function based digit systems over finite fields
Connection
Recommendations for future work

References

T. Beck, H. Brunotte, K. Scheicher, and J.M. Thuswaldner,
Number systems and tilings over Laurent series, Math. Proc.
Camb. Phil. Soc. 147 (2009) 9–29.
M.J.C. Loquias, M. Mkaouar, K. Scheicher, andJ.M. Thuswaldner, Rational digit systems over finite fields andChristol’s Theorem, J. Number Theory 171 (2017) 358–390.
S. Akiyama, C. Frougny, and J. Sakarovitch. Powers of rationals
modulo 1 and rational base number systems, Israel J. Math. 168(2008) 53–91.

Alfonso David C. Rodriguez (joint work with M. J. C. Loquias)

Fundamental Domains for P/Q-Digit Systems over Finite Fields