Minkowski’s question mark measure

Mantica, Giorgio

doi:10.1016/j.jat.2017.06.003

Cited by 1 publication

(10 citation statements)

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“…We now provide a rigorous proof of this result, which further unveils the intriguing nature of Minkowski's question mark function. The stronger conjecture that

μ

belongs to the so‐called Nevai class, also supported by numerical investigation , still lies open.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 98%

“…A result by Kinney asserts that its Hausdorff dimension can be expressed in terms of the integral of the function

{prefixlog}_{2} false(1 + x false)

with respect to the measure

μ

itself. Very precise numerical estimates of this dimension have been obtained with high precision arithmetics ; rigorous numerical lower and upper bounds derived from the Jacobi matrix of

μ

place this value between 0.874716305108207 and 0.874716305108213 . Further analytical properties of

μ

have been recently studied, among others, by the authors of .…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…In this context, Dresse and Van Assche asked whether

μ

is regular, in the sense defined below. Their numerical investigation suggested a preliminary negative answer, but their method was successively refined by a more powerful technique by the first author in , to provide compelling numerical evidence in favor of regularity of this measure. We now provide a rigorous proof of this result, which further unveils the intriguing nature of Minkowski's question mark function.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…Notwithstanding this relevance and the time‐honored history of Minkowski's question mark measure, proof of its regularity has not been achieved before. The asymptotic behavior of its orthogonal polynomials have been investigated theoretically and numerically in , with detailed pictures illustrating the abstract properties. This investigation continues in this paper from a slightly different perspective: we do not prove regularity of

μ

directly from the definition, that is, orthogonal polynomials play no rôle herein, but we use a purely measure‐theoretic criterion, which translates the idea that a regular measure is not too thin on its support.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…In this context, it is important to obtain precise estimates on the Lebesgue measure of ‘

α

–large’ cylinders. Secondly, as we will discuss momentarily, further conjectures on Minkowski's question mark measure have been formulated and numerically tested in . The rigorous proof of these conjectures should presumably require such fine control.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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Regularity of Minkowski's question mark measure, its inverse and a class of IFS invariant measures

Mantica

Тотик

2018

J. London Math. Soc.

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We prove the recent conjecture that Minkowski's question mark measure is regular in the sense of logarithmic potential theory. The proof employs: an Iterated Function System composed of Möbius maps, which yields the classical Stern–Brocot sequences, an estimate of the cardinality of large spacings between numbers in these sequences and a criterion due to Stahl and Totik. We also generalize this result to a class of balanced measures of Iterated Function Systems in one dimension.

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“…We now provide a rigorous proof of this result, which further unveils the intriguing nature of Minkowski's question mark function. The stronger conjecture that

μ

belongs to the so‐called Nevai class, also supported by numerical investigation , still lies open.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 98%

“…A result by Kinney asserts that its Hausdorff dimension can be expressed in terms of the integral of the function

{prefixlog}_{2} false(1 + x false)

with respect to the measure

μ

itself. Very precise numerical estimates of this dimension have been obtained with high precision arithmetics ; rigorous numerical lower and upper bounds derived from the Jacobi matrix of