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Sums of reciprocals of fractional parts and applications to Diophantine approximation: Part 1

Sanju Velani

Department of Mathematics University of York

Goa: 1 February 2016

Others

Joint work with Victor Beresnevich and Alan Haynes (York)

The sums of interest

We investigate the sums

SN(α, γ) := N∑

n=1

1

n‖nα− γ‖ and RN(α, γ) :=

N∑ n=1

1

‖nα− γ‖ ,

where α and γ are real parameters and ‖ · ‖ is the distance to the nearest integer. The sums are related (via partial summation):

SN(α, γ) = N∑

n=1

Rn(α, γ)

n(n + 1) +

RN(α, γ)

N + 1 .

Schmidt (1964): for any γ ∈ R and for any ε > 0

(logN)2 � SN(α, γ)� (logN)2+ε,

for almost all α ∈ R.

The sums of interest

We investigate the sums

SN(α, γ) := N∑

n=1

1

n‖nα− γ‖ and RN(α, γ) :=

N∑ n=1

1

‖nα− γ‖ ,

where α and γ are real parameters and ‖ · ‖ is the distance to the nearest integer. The sums are related (via partial summation):

SN(α, γ) = N∑

n=1

Rn(α, γ)

n(n + 1) +

RN(α, γ)

N + 1 .

Schmidt (1964): for any γ ∈ R and for any ε > 0

(logN)2 � SN(α, γ)� (logN)2+ε,

for almost all α ∈ R.

The sums of interest

Schmidt (1964): for any γ ∈ R and for any ε > 0

(logN)2 � SN(α, γ)� (logN)2+ε, (1)

for almost all α ∈ R. In the homogeneous case (γ = 0), easy to see that the ε term in (1) can be removed if α is badly approximable.

We show that when γ = 0, the l.h.s. (1) is true for all irrationals while the r.h.s. (1) is true with ε = 0 for a.a. irrationals.

More precisely:

The sums of interest

Schmidt (1964): for any γ ∈ R and for any ε > 0

(logN)2 � SN(α, γ)� (logN)2+ε, (1)

for almost all α ∈ R. In the homogeneous case (γ = 0), easy to see that the ε term in (1) can be removed if α is badly approximable.

We show that when γ = 0, the l.h.s. (1) is true for all irrationals while the r.h.s. (1) is true with ε = 0 for a.a. irrationals.

More precisely:

Homogeneous results: SN(α, 0)

Theorem. Let α ∈ R \Q. Then for N > N0

1

2 (logN)2

∀ 6 SN(α, 0) :=

N∑ n=1

1

n‖nα‖ a.a 6 34 (logN)2.

In fact, the upper bound is valid for any α := [a1, a2, . . .] such that

Ak(α) := k∑

i=1

ai = o(k 2) .

(Diamond + Vaaler: For a.a. α, for k sufficiently large Ak 6 k1+ε.)

Homogeneous results: RN(α, 0)

Theorem. Let α ∈ R \Q. Then for N > N0

N logN ∀ � RN(α, 0) :=

N∑ n=1

1

‖nα‖ .

The fact that above is valid for any irrational α is crucial for the applications in mind. (Independently: Lê + Vaaler)

Hardy + Wright: RN(α, 0)� N logN for badly approximable α. In general, not even true a.a. Indeed:

N logN log logN a.a. � RN(α, 0)

a.a. � N logN (log logN)1+� .

Now to some inhomogeneous statements.

Homogeneous results: RN(α, 0)

Theorem. Let α ∈ R \Q. Then for N > N0

N logN ∀ � RN(α, 0) :=

N∑ n=1

1

‖nα‖ .

The fact that above is valid for any irrational α is crucial for the applications in mind. (Independently: Lê + Vaaler)

Hardy + Wright: RN(α, 0)� N logN for badly approximable α. In general, not even true a.a. Indeed:

N logN log logN a.a. � RN(α, 0)

a.a. � N logN (log logN)1+� .

Now to some inhomogeneous statements.

Homogeneous results: RN(α, 0)

Theorem. Let α ∈ R \Q. Then for N > N0

N logN ∀ � RN(α, 0) :=

N∑ n=1

1

‖nα‖ .

The fact that above is valid for any irrational α is crucial for the applications in mind. (Independently: Lê + Vaaler)

Hardy + Wright: RN(α, 0)� N logN for badly approximable α. In general, not even true a.a. Indeed:

N logN log logN a.a. � RN(α, 0)

a.a. � N logN (log logN)1+� .

Now to some inhomogeneous statements.

Inhomogeneous results: a taster

Theorem. For each γ ∈ R there exists a set Aγ ⊂ R of full measure such that for all α ∈ Aγ and all sufficiently large N

SN(α, γ) := ∑

16n6N

1

n‖nα− γ‖ � (logN)2 .

The result removes the ‘epsilon’ term in Schmidt’s upper bound.

Theorem. Let α ∈ R \ (L ∪Q). Then, for all sufficiently large N and any γ ∈ R

SN(α, γ)� (logN)2 .

Schmidt’s lower bound is a.a. and depends on γ. The above is for all irrationals except possibly for Liouville numbers L.

Inhomogeneous results: a taster

Theorem. For each γ ∈ R there exists a set Aγ ⊂ R of full measure such that for all α ∈ Aγ and all sufficiently large N

SN(α, γ) := ∑

16n6N

1

n‖nα− γ‖ � (logN)2 .

The result removes the ‘epsilon’ term in Schmidt’s upper bound.

Theorem. Let α ∈ R \ (L ∪Q). Then, for all sufficiently large N and any γ ∈ R

SN(α, γ)� (logN)2 .

Schmidt’s lower bound is a.a. and depends on γ. The above is for all irrationals except possibly for Liouville numbers L.

Inhomogeneous results: a taster continued

In the lower bound result for SN(α, γ) we are not sure if we need to exclude Liouville numbers. However, it is necessary when dealing with RN(α, γ).

Theorem. Let α ∈ R \Q. Then, α 6∈ L if and only if for any γ ∈ R,

RN(α, γ) := ∑

16n6N

1

‖nα− γ‖ � N logN for N > 2.

The counting function: #{1 6 n 6 N : ‖nα− γ‖ < �}

Related to the sums, given α, γ ∈ R, N ∈ N and ε > 0, we consider the cardinality of

Nγ(α, ε) := {n ∈ N : ‖nα− γ‖ < ε, n 6 N} .

Observing that in the homogeneous case, when εN > 1, Minkowski’s Theorem for convex bodies, implies that

#N(α, ε) := #N0(α, ε) > bεNc .

Under which conditions can this bound can be reversed?

The counting function: #{1 6 n 6 N : ‖nα− γ‖ < �}

Related to the sums, given α, γ ∈ R, N ∈ N and ε > 0, we consider the cardinality of

Nγ(α, ε) := {n ∈ N : ‖nα− γ‖ < ε, n 6 N} .

Observing that in the homogeneous case, when εN > 1, Minkowski’s Theorem for convex bodies, implies that

#N(α, ε) := #N0(α, ε) > bεNc .

Under which conditions can this bound can be reversed?

The homogeneous counting results

Theorem. Let α ∈ R \Q and let (qk)k>0 be the sequence of denominators of the convergents of α. Let N ∈ N and ε > 0 such that 0 < 2ε < ‖q2α‖. Suppose that

1 2ε 6 qk 6 N

for some integer k . Then

bεNc 6 #N(α, ε) 6 32 εN . (2)

In terms of the Diophantine exponent τ(α): Let α 6∈ L ∪Q and let ν ∈ R satisfy

0 < ν < 1

τ(α) .

Then, ∃ ε0 = ε0(α) > 0 such that for any sufficiently large N and any ε with N−ν < ε < ε0, estimate (2) is satisfied.

The homogeneous counting results

Theorem. Let α ∈ R \Q and let (qk)k>0 be the sequence of denominators of the convergents of α. Let N ∈ N and ε > 0 such that 0 < 2ε < ‖q2α‖. Suppose that

1 2ε 6 qk 6 N

for some integer k . Then

bεNc 6 #N(α, ε) 6 32 εN . (2)

In terms of the Diophantine exponent τ(α): Let α 6∈ L ∪Q and let ν ∈ R satisfy

0 < ν < 1

τ(α) .

Then, ∃ ε0 = ε0(α) > 0 such that for any sufficiently large N and any ε with N−ν < ε < ε0, estimate (2) is satisfied.

The inhomogeneous counting results

Estimates for #Nγ(α, ε) are obtained from the homogenous results via the following.

Theorem. For any ε > 0 and N ∈ N, we have that

#Nγ(α, ε) 6 #N(α, 2ε) + 1.

If N ′γ(α, ε ′) 6= ∅, where N ′ := 12N and ε

′ := 12ε, then

#Nγ(α, ε) > #N ′(α, ε′) + 1 .

UPSHOT: If #{1 6 n 6 N/2 : ‖nα− γ‖ < �/2} > 0, then under the conditions of the homogeneous results

b14εNc 6 #Nγ(α, ε) 6 64 εN + 1 .

The inhomogeneous counting results

Estimates for #Nγ(α, ε) are obtained from the homogenous results via the following.

Theorem. For any ε > 0 and N ∈ N, we have that

#Nγ(α, ε) 6 #N(α, 2ε) + 1.

If N ′γ(α, ε ′) 6= ∅, where N ′ := 12N and ε

′ := 12ε, then

#Nγ(α, ε) > #N ′(α, ε′) + 1 .

UPSHOT: If #{1 6 n 6 N/2 : ‖nα− γ‖ < �/2} > 0, then under the conditions of the homogeneous results

b14εNc 6 #Nγ(α, ε) 6 64 εN + 1 .

Main Tools: Ostrowski

Ostrowski expansion of real numbers: Let α ∈ R \Q. Then, for every n ∈ N there is a unique integer K > 0 such that

qK 6 n < qK+1,

and a unique sequence {ck+1}∞k=0 of integers such that

n = ∞∑ k=0

ck+1qk , (3)

0 6 c1 < a1

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