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Multidimensional continued fractions andDiophantine exponents of lattices
UDT 2018, CIRM, Luminy
OlegN.German
Moscow State University
October 02, 2018
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 1 / 19
Irrationality exponent vs Growth of partial quotients
Irrationality exponent θ ∈ R\Q
µ(θ) = sup{γ ∈ R
∣∣∣ ∣∣θ − p/q∣∣ 6 |q|−γ admits ∞ solutions in (q, p) ∈ Z2}
Relation to the growth of partial quotientsIf
θ = [a0; a1, a2, . . .] = a0 +1
a1 +1
a2 + . . .
,pnqn
= [a0; a1, . . . , an],
thenµ(θ)− 2 = lim sup
n→+∞
ln an+1
ln qn
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 2 / 19
Irrationality exponent vs Growth of partial quotients
Irrationality exponent θ ∈ R\Q
µ(θ) = sup{γ ∈ R
∣∣∣ ∣∣θ − p/q∣∣ 6 |q|−γ admits ∞ solutions in (q, p) ∈ Z2}
Relation to the growth of partial quotientsIf
θ = [a0; a1, a2, . . .] = a0 +1
a1 +1
a2 + . . .
,pnqn
= [a0; a1, . . . , an],
thenµ(θ)− 2 = lim sup
n→+∞
ln an+1
ln qn
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 2 / 19
Geometry of continued fractions
y = θ1x
y = θ2x
K1
K2
µ(θ1)− 2 = lim supn→+∞
ln an+1
ln qnθ1 > 1, −1 < θ2 < 0
θ1 = [a0; a1, a2, . . .]
−1/θ2 = [a−1; a−2, a−3, . . .]
vn =
(qnpn
), vn = anvn−1 + vn−2
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 3 / 19
Geometry of continued fractions
y = θ1x
y = θ2x
a0
a−2
a2a1
a−1
v0
v−2
v−1v−3
K1
K2
µ(θ1)− 2 = lim supn→+∞
ln an+1
ln qnθ1 > 1, −1 < θ2 < 0
θ1 = [a0; a1, a2, . . .]
−1/θ2 = [a−1; a−2, a−3, . . .]
vn =
(qnpn
), vn = anvn−1 + vn−2
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 4 / 19
Geometry of continued fractions
y = θ1x
y = θ2x
a0
a0
a−2
a2a1
a−1
v0
v−2
v−1v−3
K1
K2
µ(θ1)− 2 = lim supn→+∞
ln an+1
ln qnθ1 > 1, −1 < θ2 < 0
θ1 = [a0; a1, a2, . . .]
−1/θ2 = [a−1; a−2, a−3, . . .]
vn =
(qnpn
), vn = anvn−1 + vn−2
an+1 = int.lenght of [vn−1,vn+1]
an+1 = int.angle α(vn) at vn
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 5 / 19
Geometry of continued fractions
y = θ1x
y = θ2x
a0
a1
a−1
a−2
a2
a0
a1
a−1a−2
v−2
v−1v−3
K1
K2
µ(θ1)− 2 = lim supn→+∞
ln an+1
ln qnθ1 > 1, −1 < θ2 < 0
θ1 = [a0; a1, a2, . . .]
−1/θ2 = [a−1; a−2, a−3, . . .]
vn =
(qnpn
), vn = anvn−1 + vn−2
an+1 = int.lenght of [vn−1,vn+1]
an+1 = int.angle α(vn) at vn
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 6 / 19
Geometry of continued fractions
y = θ1x
y = θ2x
a0
a1
a−1
a−2
a2
a0
a1
a−1a−2
v−2
v−1v−3
K1
K2
µ(θ1)− 2 = lim supn→+∞
ln an+1
ln qnθ1 > 1, −1 < θ2 < 0
θ1 = [a0; a1, a2, . . .]
−1/θ2 = [a−1; a−2, a−3, . . .]
vn =
(qnpn
), vn = anvn−1 + vn−2
an+1 = int.lenght of [vn−1,vn+1]
an+1 = int.angle α(vn) at vn
|qn| � |vn|
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 7 / 19
Geometry of continued fractions
y = θ1x
y = θ2x
a0
a1
a−1
a−2
a2
a0
a1
a−1a−2
v−2
v−1v−3
K1
K2
µ(θ1)− 2 = lim supn→+∞
ln an+1
ln qnθ1 > 1, −1 < θ2 < 0
θ1 = [a0; a1, a2, . . .]
−1/θ2 = [a−1; a−2, a−3, . . .]
vn =
(qnpn
), vn = anvn−1 + vn−2
an+1 = α(vn), |qn| � |vn|
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 8 / 19
Irrationality exponent in terms of lattice exponents
y = θ1x
y = θ2x
v0
v−2
v−1v−3
K1
K2
µ(θ1)− 2 = lim supn→+∞
ln an+1
ln qnL1(q, p) = qθ1 − p, L2(q, p) = qθ2 − p
Λ ={(L1(z), L2(z)
) ∣∣∣ z ∈ Z2}
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 9 / 19
Irrationality exponent in terms of lattice exponents
y = θ1x
y = θ2x
v0
v−2
v−1v−3
K1
K2
µ(θ1)− 2 = lim supn→+∞
ln an+1
ln qnL1(q, p) = qθ1 − p, L2(q, p) = qθ2 − p
Λ ={(L1(z), L2(z)
) ∣∣∣ z ∈ Z2}
µ(θ1) = lim supq,p∈Z|q|→∞
log(|θ1−p/q|−1
)log |q| =
= 2 + lim supq,p∈Z|q|→∞
log(|q(qθ1−p)|−1
)log |q| =
= 2 + lim supn→+∞
log(|L1(vn)·L2(vn)|−1
)log |vn|
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 10 / 19
Irrationality exponent in terms of lattice exponents
y = θ1x
y = θ2x
v0
v−2
v−1v−3
K1
K2
µ(θ1)− 2 = lim supn→+∞
ln an+1
ln qnL1(q, p) = qθ1 − p, L2(q, p) = qθ2 − p
Λ ={(L1(z), L2(z)
) ∣∣∣ z ∈ Z2}
µ(θ2) = lim supq,p∈Z|q|→∞
log(|θ2−p/q|−1
)log |q| =
= 2 + lim supq,p∈Z|q|→∞
log(|q(qθ2−p)|−1
)log |q| =
= 2 + lim supn→−∞
log(|L1(vn)·L2(vn)|−1
)log |vn|
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 11 / 19
Irrationality exponent in terms of lattice exponents
y = θ1x
y = θ2x
v0
v−2
v−1v−3
K1
K2
µ(θ1)− 2 = lim supn→+∞
ln an+1
ln qnL1(q, p) = qθ1 − p, L2(q, p) = qθ2 − p
Λ ={(L1(z), L2(z)
) ∣∣∣ z ∈ Z2}
µ(θ1) = 2 + lim supn→+∞
log(|L1(vn)·L2(vn)|−1
)log |vn|
µ(θ2) = 2 + lim supn→−∞
log(|L1(vn)·L2(vn)|−1
)log |vn|
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 12 / 19
Irrationality exponent in terms of lattice exponents
y = θ1x
y = θ2x
v0
v−2
v−1v−3
K1
K2
µ(θ1)− 2 = lim supn→+∞
ln an+1
ln qnL1(q, p) = qθ1 − p, L2(q, p) = qθ2 − p
Λ ={(L1(z), L2(z)
) ∣∣∣ z ∈ Z2}
µ(θ1) = 2 + lim supn→+∞
log(|L1(vn)·L2(vn)|−1
)log |vn|
µ(θ2) = 2 + lim supn→−∞
log(|L1(vn)·L2(vn)|−1
)log |vn|
max(µ(θ1), µ(θ2)) = 2 + 2ω(Λ)
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 13 / 19
Irrationality exponent in terms of lattice exponents
y = θ1x
y = θ2x
v0
v−2
v−1v−3
K1
K2
µ(θ1)− 2 = lim supn→+∞
ln an+1
ln qnL1(q, p) = qθ1 − p, L2(q, p) = qθ2 − p
Λ ={(L1(z), L2(z)
) ∣∣∣ z ∈ Z2}
µ(θ1) = 2 + lim supn→+∞
log(|L1(vn)·L2(vn)|−1
)log |vn|
µ(θ2) = 2 + lim supn→−∞
log(|L1(vn)·L2(vn)|−1
)log |vn|
max(µ(θ1), µ(θ2)) = 2 + 2ω(Λ)
ω(Λ) = lim supx=(x1,x2)∈Λ|x|→∞
log(|x1x2|−1/2
)log |x|
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 14 / 19
Reformulation
y = θ1x
y = θ2x
v0
v−2
v−1v−3
K1
K2
µ(θ1)− 2 = lim supn→+∞
ln an+1
ln qnL1(q, p) = qθ1 − p, L2(q, p) = qθ2 − p
Λ ={(L1(z), L2(z)
) ∣∣∣ z ∈ Z2}
ω(Λ) = lim supx=(x1,x2)∈Λ|x|→∞
log(|x1x2|−1/2
)log |x|
ω(Λ) = 12 lim sup
|v|→∞v∈V(K1)∪V(K2)
log(α(v))
log |v|
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 15 / 19
Higher dimensions: lattice exponents and Klein polyhedra
Λ a lattice in Rd, det Λ = 1
Π(x) = |x1 . . . xd|1/d for x = (x1, . . . , xd) ∈ Rd
Lattice exponent
ω(Λ) = lim supx∈Λ|x|→∞
log(Π(x)−1
)log |x|
= sup{γ ∈ R
∣∣∣Π(x) 6 |x|−γ for infinitely many x ∈ Λ}
O an orthant
Klein polyhedron
K = conv(O ∩ Λ\{0}
)
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 16 / 19
Higher dimensions: lattice exponents and Klein polyhedra
Λ a lattice in Rd, det Λ = 1
Π(x) = |x1 . . . xd|1/d for x = (x1, . . . , xd) ∈ Rd
Lattice exponent
ω(Λ) = lim supx∈Λ|x|→∞
log(Π(x)−1
)log |x|
= sup{γ ∈ R
∣∣∣Π(x) 6 |x|−γ for infinitely many x ∈ Λ}
O an orthant
Klein polyhedron
K = conv(O ∩ Λ\{0}
)OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 16 / 19
Higher dimensions: lattice exponents and Klein polyhedra
Λ a lattice in Rd, det Λ = 1
Klein polyhedron
K = conv(O ∩ Λ\{0}
)
Determinant of a face FLet v1, . . . ,vk be the vertices of F . Then
detF =∑
16i1<...<id6k
|det(vi1 , . . . ,vid)|
Determinant of an edge star StvLet r1, . . . , rk be the primitive vectors parallel to the edges incident to v. Then
detStv =∑
16i1<...<id6k
|det(ri1 , . . . , rid)|
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 17 / 19
Higher dimensions: lattice exponents and Klein polyhedra
Λ a lattice in Rd, det Λ = 1
Klein polyhedron
K = conv(O ∩ Λ\{0}
)Determinant of a face FLet v1, . . . ,vk be the vertices of F . Then
detF =∑
16i1<...<id6k
|det(vi1 , . . . ,vid)|
Determinant of an edge star StvLet r1, . . . , rk be the primitive vectors parallel to the edges incident to v. Then
detStv =∑
16i1<...<id6k
|det(ri1 , . . . , rid)|
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 17 / 19
Higher dimensions: lattice exponents and Klein polyhedra
Λ a lattice in Rd, K1, . . . ,K2d its Klein polyhedra, V(Ki) set of vertices
Two-dimensional statement
ω(Λ) = 12 lim sup
|v|→∞v∈V(K1)∪V(K2)
log(detStv)
log |v|
Question, arbitrary dimension
Is it true that ω(Λ) � lim sup|v|→∞
v∈⋃
i V(Ki)
log(detStv)
log |v|?
E.Bigushev, O.G., 2018
For d = 3 we have ω(Λ) 6 23 lim sup|v|→∞
v∈⋃
i V(Ki)
log(detStv)
log |v|
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 18 / 19
Higher dimensions: lattice exponents and Klein polyhedra
Λ a lattice in Rd, K1, . . . ,K2d its Klein polyhedra, V(Ki) set of vertices
Two-dimensional statement
ω(Λ) = 12 lim sup
|v|→∞v∈V(K1)∪V(K2)
log(detStv)
log |v|
Question, arbitrary dimension
Is it true that ω(Λ) � lim sup|v|→∞
v∈⋃
i V(Ki)
log(detStv)
log |v|?
E.Bigushev, O.G., 2018
For d = 3 we have ω(Λ) 6 23 lim sup|v|→∞
v∈⋃
i V(Ki)
log(detStv)
log |v|
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 18 / 19
Higher dimensions: lattice exponents and Klein polyhedra
Λ a lattice in Rd, K1, . . . ,K2d its Klein polyhedra, V(Ki) set of vertices
Two-dimensional statement
ω(Λ) = 12 lim sup
|v|→∞v∈V(K1)∪V(K2)
log(detStv)
log |v|
Question, arbitrary dimension
Is it true that ω(Λ) � lim sup|v|→∞
v∈⋃
i V(Ki)
log(detStv)
log |v|?
E.Bigushev, O.G., 2018
For d = 3 we have ω(Λ) 6 23 lim sup|v|→∞
v∈⋃
i V(Ki)
log(detStv)
log |v|
OlegN.German Multidimensional continued fractions and Diophantine exponents of lattices 18 / 19