On the theorem of Davenport and generalized Dedekind sums

Borda, Bence

doi:10.1016/j.jnt.2016.08.016

Cited by 7 publications

(12 citation statements)

References 5 publications

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“…We proved the same result for S(α, N ) with the slightly worse error term O(log log N ) in a previous paper [9]. In contrast to A(α) and Λ(α), there seems to be no simple way to compute the value of c(α) directly from the continued fraction expansion.…”

Section: Introductionsupporting

confidence: 77%

“…This method goes back to Davenport [11], and more recently has also been used in [5,6,7,16,21]. We follow the steps in our previous paper [9], where we considered irrationals whose sequence of partial quotients is reasonably well-behaved (e.g. bounded, or increasing at a regular rate such as for Euler's number).…”

Section: Discrepancy Via the Parseval Formula 21 The Main Estimatesmentioning

confidence: 99%

“…Finally, we will need two different evaluations of the Diophantine sum appearing in Proposition 7. On the one hand, for general α we have [10, p. 110], [9]…”

Section: Discrepancy Via the Parseval Formula 21 The Main Estimatesmentioning

confidence: 99%

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Optimal and typical $L^2$ discrepancy of 2-dimensional lattices

Borda¹

2021

Preprint

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We undertake a detailed study of the L 2 discrepancy of rational and irrational 2-dimensional lattices either with or without symmetrization. We give a full characterization of lattices with optimal L 2 discrepancy in terms of the continued fraction partial quotients, and compute the precise asymptotics whenever the continued fraction expansion is explicitly known, such as for quadratic irrationals or Euler's number e. In the metric theory, we find the asymptotics of the L 2 discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices.

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Section: Introductionsupporting

confidence: 77%

Section: Discrepancy Via the Parseval Formula 21 The Main Estimatesmentioning

confidence: 99%

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Optimal and typical $L^2$ discrepancy of 2-dimensional lattices

Borda¹

2021

Preprint

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“…A few years later the constant involved on the right hand side of (8) was identified to have the explicit expression 4 15 √ 5 (see [5]). Furthermore, it is well known that log F n n. This shows that all the L 2 discrepancies under study of the Fibonacci lattice are of optimal order of magnitude with respect to the corresponding Roth-type lower bounds.…”

Section: Fnmentioning

confidence: 99%

Extreme and periodic $L_2$ discrepancy of plane point sets

Hinrichs

Kritzinger²,

Pillichshammer³

2021

Acta Arith.

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We study several discrepancy notions of two wellknown instances of plane point sets, namely the Hammersley point set and rational lattices. The discrepancies are considered with respect to the L 2 norm and a variety of test sets. We define (standard) L 2 discrepancy, extreme L 2 discrepancy and periodic L 2 discrepancy. Let P = {x 0 , x 1 ,. .. , x N −1 } be an arbitrary N-element point set in the unit square [0, 1) 2. For any measurable subset B of [0, 1] 2 we define the counting function A(B, P) := n ∈ {0, 1,. .. , N − 1} : x n ∈ B , i.e., the number of elements from P that belong to the set B. By the local discrepancy of P with respect to a given measurable "test set" B one understands the expression A(B, P) − N λ(B), where λ denotes the Lebesgue measure of B. A global discrepancy measure is then obtained by considering a norm of the local discrepancy with respect to a fixed class of test sets. Here we restrict ourselves to the L 2 norm, but we vary the class of test sets. The (standard) L 2 discrepancy uses the class of axis-parallel squares anchored at the origin as test sets. The formal definition is L 2,N (P) :=

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“…The last inequality follows from a classical method based on the pigeonhole principle, see e.g. [9] for a detailed proof. Therefore…”

Section: A Shifted Cotangent Summentioning

confidence: 99%

A conjecture of Zagier and the value distribution of quantum modular forms

Aistleitner¹,

Borda²

2021

Preprint

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In his influential paper on quantum modular forms, Zagier developed a conjectural framework describing the behavior of certain quantum knot invariants under the action of the modular group on their argument. More precisely, when J K,0 denotes the colored Jones polynomial of a knot K, Zagier's modularity conjecture describes the asymptotics of the quotient J K,0 (e 2πiγ(x) )/J K,0 (e 2πix ) as x → ∞ along rationals with bounded denominators, where γ ∈ SL(2, Z). This problem is most accessible for the figure-eight knot 4 1 , when the colored Jones polynomial has an explicit expression in terms of the q-Pochhammer symbol. Zagier also conjectured that the function h(x) = log(J 41,0 (e 2πix )/J 41,0 (e 2πi/x )) can be extended to a function on R which is continuous at irrationals. In the present paper, we prove Zagier's continuity conjecture for all irrationals which have unbounded partial quotients in their continued fraction expansion. In particular, the continuity conjecture holds almost everywhere on the real line. We also establish a smooth approximation of h, uniform over all rationals, in accordance with the modularity conjecture. As an application, we find the limit distribution (after a suitable centering and rescaling) of log(J 41,0 (e 2πix )), when x ranges over all reduced rationals in (0, 1) with denominator at most N , as N → ∞, thereby confirming a conjecture of Bettin and Drappeau.

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On the theorem of Davenport and generalized Dedekind sums

Cited by 7 publications

References 5 publications

Optimal and typical $L^2$ discrepancy of 2-dimensional lattices

Optimal and typical $L^2$ discrepancy of 2-dimensional lattices

Extreme and periodic $L_2$ discrepancy of plane point sets

A conjecture of Zagier and the value distribution of quantum modular forms

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