Innite families of number systems
Jakub Krásenský,Czech Technical University (Prague),

Attila Kovács

Eötvös Loránd University (Budapest)

Numeration 2018,
Paris

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Outline1

GNSs in lattices

2Innitely many GNSs with the same radix

3

Sparse alphabets

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Number system (GNS)dA lattice (Z );d×d ;a radix L ∈ Z

d containing zero.

an alphabet A ⊂ Z
Denitiond if every element

The pair (L, A) is a number system (GNS) in Z

x ∈ Zd has a unique representation of the formx=

NX

Lk ak ,ak ∈ A.

k=0
Note: The binary and the decimal system are not GNS since they
do not allow to represent negative numbers. (−2, {0, 1}) is a GNS
in Z and so is (3, {−1, 0, 1}).

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Number system (GNS)dA lattice (Z );d×d ;a radix L ∈ Z

d containing zero.

an alphabet A ⊂ Z
Denitiond if every element

The pair (L, A) is a number system (GNS) in Z

x ∈ Zd has a unique representation of the formx=

NX

Lk ak ,ak ∈ A.

k=0
Note: The binary and the decimal system are not GNS since they
do not allow to represent negative numbers. (−2, {0, 1}) is a GNS
in Z and so is (3, {−1, 0, 1}).

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Number system (GNS)dA lattice (Z );d×d ;a radix L ∈ Z

d containing zero.

an alphabet A ⊂ Z
Denitiond if every element

The pair (L, A) is a number system (GNS) in Z

x ∈ Zd has a unique representation of the formx=

NX

Lk ak ,ak ∈ A.

k=0
Note: The binary and the decimal system are not GNS since they
do not allow to represent negative numbers. (−2, {0, 1}) is a GNS
in Z and so is (3, {−1, 0, 1}).

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Necessary conditions−1 ) < 1 (Vince, 1993),

The radix L must be expansive, i.e. ρ(Lthe alphabet A must be a complete residue system modulo L,
det(L − I ) 6= ±1 (the unit condition).

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Classical question: Given a radix L ∈ Zd×d , does there exist analphabet such that (L, A) is a GNS?
Theorem (Steidl, 1989; Kátai, 1994)

Let β ∈ Z[i]. Then there exists an alphabet A such that (β, A) is a
GNS if and only if |β| =6 1, |β − 1| =6 1. The same holds in OK
where K is any imaginary quadratic eld.
The used digits lie in a parallelogram around the origin.
Theorem (Germán, Kovács, 2007)

If ρ(L−1 ) < 1/2, then there always exists an alphabet such that
(L, A) is a GNS.
They use the dense alphabet, i.e. the smallest representative (in acertain vector norm) from every congruence class.

−1 ) < 1 is a necessary condition.

Reminder: ρ(LJ. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Classical question: Given a radix L ∈ Zd×d , does there exist analphabet such that (L, A) is a GNS?
Theorem (Steidl, 1989; Kátai, 1994)

Let β ∈ Z[i]. Then there exists an alphabet A such that (β, A) is a
GNS if and only if |β| =6 1, |β − 1| =6 1. The same holds in OK
where K is any imaginary quadratic eld.
The used digits lie in a parallelogram around the origin.
Theorem (Germán, Kovács, 2007)

If ρ(L−1 ) < 1/2, then there always exists an alphabet such that
(L, A) is a GNS.
They use the dense alphabet, i.e. the smallest representative (in acertain vector norm) from every congruence class.

−1 ) < 1 is a necessary condition.

Reminder: ρ(LJ. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Classical question: Given a radix L ∈ Zd×d , does there exist analphabet such that (L, A) is a GNS?
Theorem (Steidl, 1989; Kátai, 1994)

Let β ∈ Z[i]. Then there exists an alphabet A such that (β, A) is a
GNS if and only if |β| =6 1, |β − 1| =6 1. The same holds in OK
where K is any imaginary quadratic eld.
The used digits lie in a parallelogram around the origin.
Theorem (Germán, Kovács, 2007)

If ρ(L−1 ) < 1/2, then there always exists an alphabet such that
(L, A) is a GNS.
They use the dense alphabet, i.e. the smallest representative (in acertain vector norm) from every congruence class.

−1 ) < 1 is a necessary condition.

Reminder: ρ(LJ. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Classical question: Given a radix L ∈ Zd×d , does there exist analphabet such that (L, A) is a GNS?
Theorem (Steidl, 1989; Kátai, 1994)

Let β ∈ Z[i]. Then there exists an alphabet A such that (β, A) is a
GNS if and only if |β| =6 1, |β − 1| =6 1. The same holds in OK
where K is any imaginary quadratic eld.
The used digits lie in a parallelogram around the origin.
Theorem (Germán, Kovács, 2007)

If ρ(L−1 ) < 1/2, then there always exists an alphabet such that
(L, A) is a GNS.
They use the dense alphabet, i.e. the smallest representative (in acertain vector norm) from every congruence class.

−1 ) < 1 is a necessary condition.

Reminder: ρ(LJ. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Classical question: Given a radix L ∈ Zd×d , does there exist analphabet such that (L, A) is a GNS?
Theorem (Steidl, 1989; Kátai, 1994)

Let β ∈ Z[i]. Then there exists an alphabet A such that (β, A) is a
GNS if and only if |β| =6 1, |β − 1| =6 1. The same holds in OK
where K is any imaginary quadratic eld.
The used digits lie in a parallelogram around the origin.
Theorem (Germán, Kovács, 2007)

If ρ(L−1 ) < 1/2, then there always exists an alphabet such that
(L, A) is a GNS.
They use the dense alphabet, i.e. the smallest representative (in acertain vector norm) from every congruence class.

−1 ) < 1 is a necessary condition.

Reminder: ρ(LJ. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Question 1: Given a radix L ∈ Zd×d , how many alphabets for thisradix do exist?
So far known:
Radix is not expansive or the unit condition fails=⇒ there isno such alphabet.
For −2 in Z, only the alphabets {0, 1} and {0, −1} aresuitable.
Similarly, for −1 + i in Z[i] there are only four good alphabets.
Matula, 1982: In Z, for every β with |β| ≥ 3 there areinnitely many alphabets.
Block diagonal radices



L1 0
0 L2



are easy to handle. (Thesublattices can be represented independently  Kovács, 2014
and also Indlekofer, Kátai, Racskó, 1993, etc.)

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Question 1: Given a radix L ∈ Zd×d , how many alphabets for thisradix do exist?
So far known:
Radix is not expansive or the unit condition fails=⇒ there isno such alphabet.
For −2 in Z, only the alphabets {0, 1} and {0, −1} aresuitable.
Similarly, for −1 + i in Z[i] there are only four good alphabets.
Matula, 1982: In Z, for every β with |β| ≥ 3 there areinnitely many alphabets.
Block diagonal radices



L1 0
0 L2



are easy to handle. (Thesublattices can be represented independently  Kovács, 2014
and also Indlekofer, Kátai, Racskó, 1993, etc.)

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Question 1: Given a radix L ∈ Zd×d , how many alphabets for thisradix do exist?
So far known:
Radix is not expansive or the unit condition fails=⇒ there isno such alphabet.
For −2 in Z, only the alphabets {0, 1} and {0, −1} aresuitable.
Similarly, for −1 + i in Z[i] there are only four good alphabets.
Matula, 1982: In Z, for every β with |β| ≥ 3 there areinnitely many alphabets.
Block diagonal radices



L1 0
0 L2



are easy to handle. (Thesublattices can be represented independently  Kovács, 2014
and also Indlekofer, Kátai, Racskó, 1993, etc.)

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Question 1: Given a radix L ∈ Zd×d , how many alphabets for thisradix do exist?
So far known:
Radix is not expansive or the unit condition fails=⇒ there isno such alphabet.
For −2 in Z, only the alphabets {0, 1} and {0, −1} aresuitable.
Similarly, for −1 + i in Z[i] there are only four good alphabets.
Matula, 1982: In Z, for every β with |β| ≥ 3 there areinnitely many alphabets.
Block diagonal radices



L1 0
0 L2



are easy to handle. (Thesublattices can be represented independently  Kovács, 2014
and also Indlekofer, Kátai, Racskó, 1993, etc.)

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Question 1: Given a radix L ∈ Zd×d , how many alphabets for thisradix do exist?
So far known:
Radix is not expansive or the unit condition fails=⇒ there isno such alphabet.
For −2 in Z, only the alphabets {0, 1} and {0, −1} aresuitable.
Similarly, for −1 + i in Z[i] there are only four good alphabets.
Matula, 1982: In Z, for every β with |β| ≥ 3 there areinnitely many alphabets.
Block diagonal radices



L1 0
0 L2



are easy to handle. (Thesublattices can be represented independently  Kovács, 2014
and also Indlekofer, Kátai, Racskó, 1993, etc.)

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Question 1: Given a radix L ∈ Zd×d , how many alphabets for thisradix do exist?
So far known:
Radix is not expansive or the unit condition fails=⇒ there isno such alphabet.
For −2 in Z, only the alphabets {0, 1} and {0, −1} aresuitable.
Similarly, for −1 + i in Z[i] there are only four good alphabets.
Matula, 1982: In Z, for every β with |β| ≥ 3 there areinnitely many alphabets.
Block diagonal radices



L1 0
0 L2



are easy to handle. (Thesublattices can be represented independently  Kovács, 2014
and also Indlekofer, Kátai, Racskó, 1993, etc.)

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets
Our result:
Theorem

Let L be an operator on a d -dimensional lattice satisfying
ρ(L−1 ) ≤ 1/2 for which 2 is not an eigenvalue. There always existinnitely many GNSs with radix L except for the case d ≤ 2.whered = 2 and L has complex eigenvalues (where we do not know), andfor the case of radix −2 in Z, where only two GNSs exist.
Note: Except for dimension d = 2, the assumptions are weaker
than the assumptions of the theorem of Germán and Kovács.
One of the main ingredients:
Proposition
If L is block-triangular, L =





L1 C
, and for L1 and L2 there exists
0 L2

one GNS, then for L there exist innitely many GNSs.
J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets
Our result:
Theorem

Let L be an operator on a d -dimensional lattice satisfying
ρ(L−1 ) ≤ 1/2 for which 2 is not an eigenvalue. There always existinnitely many GNSs with radix L except for the case d ≤ 2.whered = 2 and L has complex eigenvalues (where we do not know), andfor the case of radix −2 in Z, where only two GNSs exist.
Note: Except for dimension d = 2, the assumptions are weaker
than the assumptions of the theorem of Germán and Kovács.
One of the main ingredients:
Proposition
If L is block-triangular, L =





L1 C
, and for L1 and L2 there exists
0 L2

one GNS, then for L there exist innitely many GNSs.
J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets
Our result:
Theorem

Let L be an operator on a d -dimensional lattice satisfying
ρ(L−1 ) ≤ 1/2 for which 2 is not an eigenvalue. There always existinnitely many GNSs with radix L except for the case d ≤ 2.whered = 2 and L has complex eigenvalues (where we do not know), andfor the case of radix −2 in Z, where only two GNSs exist.
Note: Except for dimension d = 2, the assumptions are weaker
than the assumptions of the theorem of Germán and Kovács.
One of the main ingredients:
Proposition
If L is block-triangular, L =





L1 C
, and for L1 and L2 there exists
0 L2

one GNS, then for L there exist innitely many GNSs.
J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets
Our result:
Theorem

Let L be an operator on a d -dimensional lattice satisfying
ρ(L−1 ) ≤ 1/2 for which 2 is not an eigenvalue. There always existinnitely many GNSs with radix L except for the case d ≤ 2.whered = 2 and L has complex eigenvalues (where we do not know), andfor the case of radix −2 in Z, where only two GNSs exist.
Note: Except for dimension d = 2, the assumptions are weaker
than the assumptions of the theorem of Germán and Kovács.
One of the main ingredients:
Proposition
If L is block-triangular, L =





L1 C
, and for L1 and L2 there exists
0 L2

one GNS, then for L there exist innitely many GNSs.
J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets
Our result:
Theorem

Let L be an operator on a d -dimensional lattice satisfying
ρ(L−1 ) ≤ 1/2 for which 2 is not an eigenvalue. There always existinnitely many GNSs with radix L except for the case where d = 2
and L has complex eigenvalues (where we do not know), and forthe case of radix −2 in Z, where only two GNSs exist.
Note: Except for dimension d = 2, the assumptions are weaker
than the assumptions of the theorem of Germán and Kovács.
One of the main ingredients:
Proposition
If L is block-triangular, L =





L1 C
, and for L1 and L2 there exists
0 L2

one GNS, then for L there exist innitely many GNSs.
J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets
Our result:
Theorem

Let L be an operator on a d -dimensional lattice satisfying
ρ(L−1 ) ≤ 1/2 for which 2 is not an eigenvalue. There always existinnitely many GNSs with radix L except for the case where d = 2
and L has complex eigenvalues (where we do not know), and forthe case of radix −2 in Z, where only two GNSs exist.
Note: Except for dimension d = 2, the assumptions are weaker
than the assumptions of the theorem of Germán and Kovács.
One of the main ingredients:
Proposition
If L is block-triangular, L =





L1 C
, and for L1 and L2 there exists
0 L2

one GNS, then for L there exist innitely many GNSs.
J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets
Our result:
Theorem

Let L be an operator on a d -dimensional lattice satisfying
ρ(L−1 ) ≤ 1/2 for which 2 is not an eigenvalue. There always existinnitely many GNSs with radix L except for the case where d = 2
and L has complex eigenvalues (where we do not know), and forthe case of radix −2 in Z, where only two GNSs exist.
Note: Except for dimension d = 2, the assumptions are weaker
than the assumptions of the theorem of Germán and Kovács.
One of the main ingredients:
Proposition
If L is block-triangular, L =





L1 C
, and for L1 and L2 there exists
0 L2

one GNS, then for L there exist innitely many GNSs.
J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Sparse alphabets
Question 2: Can all the digits be far away from the origin?
Denition

d×d , a sequence of alphabets (A )n n∈N is called

Given a radix L ∈ Z

a family of arbitrarily sparse alphabets if for any given ball B
around the origin, there exists an nB such that for n ≥ nB we haveAn ∩ B = {0}.
dEquivalently we can require that for any nite 0 ∈/ S ⊂ Z , thealphabets An do not use any digits from S for n ≥ nS .
If (L, An ) is a GNS for every n, we have a family of arbitrarily

sparse GNSs.
J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Sparse alphabets
Question 2: Can all the digits be far away from the origin?
Denition

d×d , a sequence of alphabets (A )n n∈N is called

Given a radix L ∈ Z

a family of arbitrarily sparse alphabets if for any given ball B
around the origin, there exists an nB such that for n ≥ nB we haveAn ∩ B = {0}.
dEquivalently we can require that for any nite 0 ∈/ S ⊂ Z , thealphabets An do not use any digits from S for n ≥ nS .
If (L, An ) is a GNS for every n, we have a family of arbitrarily

sparse GNSs.
J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Sparse alphabets
Question 2: Can all the digits be far away from the origin?
Denition

d×d , a sequence of alphabets (A )n n∈N is called

Given a radix L ∈ Z

a family of arbitrarily sparse alphabets if for any given ball B
around the origin, there exists an nB such that for n ≥ nB we haveAn ∩ B = {0}.
dEquivalently we can require that for any nite 0 ∈/ S ⊂ Z , thealphabets An do not use any digits from S for n ≥ nS .
If (L, An ) is a GNS for every n, we have a family of arbitrarily

sparse GNSs.
J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Sparse alphabets
Question 2: Can all the digits be far away from the origin?
Denition

d×d , a sequence of alphabets (A )n n∈N is called

Given a radix L ∈ Z

a family of arbitrarily sparse alphabets if for any given ball B
around the origin, there exists an nB such that for n ≥ nB we haveAn ∩ B = {0}.
dEquivalently we can require that for any nite 0 ∈/ S ⊂ Z , thealphabets An do not use any digits from S for n ≥ nS .
If (L, An ) is a GNS for every n, we have a family of arbitrarily

sparse GNSs.
J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

The nal result:
TheoremSuppose that ρ(L−1 ) ≤ 1/2 and 2 is not an eigenvalue of L. Then
there exists a family of arbitrarily sparse GNSs except for the casewhen every eigenvalue of L is either an integer or a non-real
algebraic number of degree 2, and has geometric multiplicity 1.
Problems arise only if all factors of the characteristic polynomial
have degree less than 3.

Again, part of the proof uses the block-triangularisable matrices.
But we need the building stones.

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

The nal result:
TheoremSuppose that ρ(L−1 ) ≤ 1/2 and 2 is not an eigenvalue of L. Then
there exists a family of arbitrarily sparse GNSs except for the casewhen every eigenvalue of L is either an integer or a non-real
algebraic number of degree 2, and has geometric multiplicity 1.
Problems arise only if all factors of the characteristic polynomial
have degree less than 3.

Again, part of the proof uses the block-triangularisable matrices.
But we need the building stones.

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

The nal result:
TheoremSuppose that ρ(L−1 ) ≤ 1/2 and 2 is not an eigenvalue of L. Then
there exists a family of arbitrarily sparse GNSs except for the casewhen every eigenvalue of L is either an integer or a non-real
algebraic number of degree 2, and has geometric multiplicity 1.
Problems arise only if all factors of the characteristic polynomial
have degree less than 3.

Again, part of the proof uses the block-triangularisable matrices.
But we need the building stones.

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets
The dicult part:
Theorem

Let L ∈ Zd×d such that
ρ(L−1 ) < 1/2, i.e. all eigenvalues of L are bigger than 2;
L is diagonalizable over C;
the characteristic polynomial of L is f k for some k ∈ N and f
is irreducible over Q;
either d ≥ 3 or L has real eigenvalues and d = 2.
Then there exists a family of arbitrarily sparse GNSs with radix L.
It is easy to construct a sequence of vector norms k · kn such thatthe k · kn -dense alphabet An gives a GNS for every n. The problem
is that quite often all these alphabets are in fact the same.

J. Krásenský, A. Kovács

Innite families of number systems

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets
The dicult part:
Theorem

Let L ∈ Zd×d such that
ρ(L−1 ) < 1/2, i.e. all eigenvalues of L are bigger than 2;
L is diagonalizable over C;
the characteristic polynomial of L is f k for some k ∈ N and f
is irreducible over Q;
either d ≥ 3 or L has real eigenvalues and d = 2.
Then there exists a family of arbitrarily sparse GNSs with radix L.
It is easy to construct a sequence of vector norms k · kn such thatthe k · kn -dense alphabet An gives a GNS for every n. The problem
is that quite often all these alphabets are in fact the same.

J. Krásenský, A. Kovács

Innite families of number systems

Suppose a regular expansive matrix L ∈ Zd×d is diagonalisable over C. Thenthe following statements are equivalent:(A) For a basis B let us denote by An the dense alphabet with respect to then)vector norm dened as kxk(BD:= (BDn )−1 x ∞ where∞
Dn := diag(1, n, . . . , n). With this notation, the eigenbasis B of L can bechosen so that the sequence of alphabets An forms a family of arbitrarily
sparse alphabets with respect to L.
(B) The eigenbasis B can be chosen so that for every congruence class
Ta := LZd + a except for T0 the expression |[B −1 z]1 | does not attain aminimum on Ta (it only has an inmum).
(C) There exists an eigenvector b of LT such that for every congruence class
Ta := LZd + a except for T0 the expression |b T z| does not attain aminimum on Ta (it only has an inmum).
(D) There exists an eigenvector b of LT such that if bT z = 0 for z ∈ Zd , thenz ≡ 0 modulo L. Further, the set {b T z : z ∈ Zd } is either dense in C or adense subset of R.
(E) There exists an eigenvector b of LT whose coordinates are linearly
independent over Q. Further, either d ≥ 3 or d = 2 and L has only real
eigenvalues.
(F) The characteristic polynomial of L is in the form f k where f is some
irreducible integer polynomial. Further, either d ≥ 3 or d = 2 and L has
only real eigenvalues.

Starting point:
Theorem (Germán, Kovács, 2007)

If ρ(L−1 ) < 1/2, then a GNS always exists.
Results:
TheoremSuppose that ρ(L−1 ) ≤ 1/2 and 2 is not an eigenvalue of L. There
always exist innitely many GNSs with radix L except for the casewhere d = 2 and L has complex eigenvalues (where we do notknow), and the case of radix −2 in Z, where only two GNSs exist.
TheoremSuppose that ρ(L−1 ) ≤ 1/2 and 2 is not an eigenvalue of L. Then
there exists a family of arbitrarily sparse GNSs except for the casewhen every eigenvalue of L is either an integer or a non-real
algebraic number of degree 2, and has geometric multiplicity 1.

GNSs in lattices
Innitely many GNSs with the same radix
Sparse alphabets

Thank you for your attention!

J. Krásenský, A. Kovács

Innite families of number systems