Points of bounded height on oscillatory sets

Comte, G.; Miller, Chris

doi:10.1093/qmath/hax021

Cited by 5 publications

(6 citation statements)

References 26 publications

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“…where d is a bound for the degree of f (k/D N + i/24) over Q and c is effective. Theorem 1.3 now follows from (13).…”

Section: Dynamics Over Cmentioning

confidence: 91%

“…One possibility is to assume that f satisfies some nice form of differential equation. In this direction, see work by Pila [31,33], Thomas and the second author [20,21], Binyamini and Novikov [4] and Comte and Miller [13]. In a different direction, Masser [26] proved a very precise zero estimate for the Riemann zeta function and used it to show that a c(log H) 2 (log log H) −2 bound holds for the restriction of the zeta function to (2,3).…”

Section: Introductionmentioning

confidence: 99%

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Rational values of transcendental functions and arithmetic dynamics

Boxall¹,

Jones²,

Schmidt³

2018

Preprint

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We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work with p-adic methods to obtain, for each positive ε, an upper bound of the form cD 3n/4+εn on the number of irreducible factors of P •n (X) − P •n (α) over K, where K is a number field, P is a polynomial of degree D ≥ 2 over K, P •n is the n-th iterate of P , α is a point in K for which {P •n (α) : n ∈ N} is infinite and c depends effectively on P, α, [K : Q] and ε.

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“…where d is a bound for the degree of f (k/D N + i/24) over Q and c is effective. Theorem 1.3 now follows from (13).…”

Section: Dynamics Over Cmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Rational values of transcendental functions and arithmetic dynamics

Boxall¹,

Jones²,

Schmidt³

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…One possibility is to assume that f satisfies some nice form of differential equation. In this direction, see work by Pila [26,27], Thomas and the second author [17,18], Binyamini and Novikov [5] and Comte and Miller [11]. In a different direction, Masser [21] proved a very precise zero estimate for the Riemann zeta function and used it to show that a c.log H / 2 .log log H / 2 bound holds for the restriction of the zeta function to .2; 3/.…”

Section: Introductionmentioning

confidence: 99%

Rational values of transcendental functions and arithmetic dynamics

2021

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We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work with p-adic methods to obtain, for each positive ", an upper bound of the form cD 3n=4C"n on the number of irreducible factors of P ın .X / P ın .˛/ over K, where K is a number field, P is a polynomial of degree D 2 over K, P ın is the n-th iterate of P , ˛is a point in K for which ¹P ın .˛/ W n 2 Nº is infinite and c depends effectively on P; ˛; OEK W Q and ".

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“…Further examples of this sort can be found in [8] and [14]. Certain point counting theorems in the second category have recently appeared in [7]. In both categories, sharp cone decomposition theorems are by now at our disposal ( [14] and [23]), in analogy with the cell decomposition theorem known for o-minimal structures.…”

Section: Introductionmentioning

confidence: 99%

Counting algebraic points in expansions of o-minimal structures by a dense set

Eleftheriou¹

2017

Preprint

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The Pila-Wilkie theorem states that if a set X ⊆ R n is definable in an o-minimal structure R and contains 'many' rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion R = R, P of R by a dense set P , which is either an elementary substructure of R, or it is independent, as follows. If X is definable in R and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is ∅-definable in R, P , where R is the real field.

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Points of bounded height on oscillatory sets

Cited by 5 publications

References 26 publications

Rational values of transcendental functions and arithmetic dynamics

Rational values of transcendental functions and arithmetic dynamics

Rational values of transcendental functions and arithmetic dynamics

Counting algebraic points in expansions of o-minimal structures by a dense set

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