Η Εικασία Duffin-Schaeffer

Φουντουλάκης, Στυλιανός; Fountoulakis, Stylianos

dc.contributor.author	Φουντουλάκης, Στυλιανός	el
dc.contributor.author	Fountoulakis, Stylianos	en
dc.date.accessioned	2025-12-01T09:20:59Z
dc.date.available	2025-12-01T09:20:59Z
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/62955
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.30651
dc.rights	Default License
dc.subject	Mέτρο	el
dc.subject	Προσέγγιση	el
dc.subject	Ρητοί	el
dc.subject	Φράγμα	el
dc.subject	Σχεδόν όλοι	el
dc.subject	Gallagher's law	en
dc.subject	Septuple	en
dc.subject	Multiplicative data	en
dc.subject	Graphs	en
dc.subject	Quality	en
dc.subject	Density	en
dc.title	Η Εικασία Duffin-Schaeffer	el
heal.type	bachelorThesis
heal.generalDescription	Θα ήθελα να ευχαριστήσω ιδιαιτέρως τον κύριο Γιαννόπουλο για την αμέριστη βοήθεια που μου προσέφερε δίχως την ύπαρξη της οποίας δε θα είχα ολοκληρώσει την εκπόνηση της διπλωματικής μου εργασίας.	el
heal.classification	Θεωρία Αριθμών	el
heal.classification	Μαθηματική Ανάλυση	el
heal.language	el
heal.access	free
heal.recordProvider	ntua	el
heal.publicationDate	2025-02-21
heal.abstract	Let ψ : N → R≥0 be an arbitrary function from the positive integers to the non-negative reals. Consider the set A of real numbers x ∈ [0, 1] for which there are infinitely many reduced fractions a/q such that \|x − a/q\| < ψ(q)/q. Using the first Borel-Cantelli lemma, one can easily check that if ∑ψ(q)ϕ(q)/q < ∞ then A has Lebesgue measure λ(A) = 0. The Duffin-Schaeffer conjecture asserts that if ∑ψ(q)ϕ(q)/q = ∞ then A has Lebesgue measure λ(A) = 1. In this thesis we present the work of D. Koukoulopoulos and J. Maynard who confirmed this conjecture in full generality. Corollaries of the main result are: (i) a conjecture due to Catlin regarding non-reduced solutions to the inequality \|x − a/q\| < ψ(q)/q, which refines Khinchin’s theorem, and (ii) a Hausdorff measure generalization of the Duffin-Schaeffer conjecture. The former was already known to follow straightforwardly from the Duffin-Schaeffer conjecture. The latter was already known to follow from the Duffin-Schaeffer conjecture via a mass transference principle of Beresnevich and Velani. A brief description of the strategy of the proof starts with a theorem of Gallagher who had established the abstract zero-one law λ(A) ∈ {0, 1} via an ergodic argument. Using this fact and a deterministic variant of the second Borel-Cantelli lemma, one can reduce the problem to showing that λ(Aq ∩ Ar) << λ(Aq)λ(Ar) in a suitably averaged sense, where Aq = ⋃ [ a − ψ(q)/ q , a + ψ(q)/ q ] , where gcd(a,q)=1 and a runs from 1 to q. This idea is known as quasi-independence on average, and in fact it was used by Duffin and Schaeffer who had settled their conjecture in certain arithmetically structured cases. Further important steps were then made by Erdős and Vaaler, who solved the case ψ(q) << 1/q. They measured the overlaps of the sets Aq and showed that if q , r then λ(Aq ∩ Ar) /λ(Aq)λ(Ar) << ∏ ( 1 + 1/p ) = F(n, t), where n = n(q, r) = qr/ gcd(q, r)^2 and t = t(q, r) = max{rψ(q), qψ(r)} /gcd(q, r) and p\|n, p>t. If we do not have quasi-independence on average, then F(n, t) can be is large. Thus, the question is for which sets S , of a given density in a given discrete interval, there can be many pairs (q, r) ∈ S^2 such that F(n(q, r), t(q, r)) is large. The new idea of Koukoulopoulos and Maynard is to construct a weighted graph that encodes this information, and to use additive combinatorics to determine the structure of such a set. The final conclusion is that there must be a fixed, large number that divides a positive proportion of the elements of S . To prove this, they adapt Roth’s density increment strategy, keeping control on the density and arithmetic data of the evolving weighted graph. When a large divisor is found, it can be “factored out", and the problem more or less reduces to the case that can be treated with the methods developed by Erdős and Vaaler.	en
heal.advisorName	Γιαννόπουλος, Απόστολος	el
heal.committeeMemberName	Γιαννόπουλος, Απόστολος	el
heal.committeeMemberName	Γρηγοριάδης, Βασίλειος	el
heal.committeeMemberName	Κανελλόπουλος, Βασίλειος	el
heal.academicPublisher	Εθνικό Μετσόβιο Πολυτεχνείο. Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών. Τομέας Μαθηματικών	el
heal.academicPublisherID	ntua
heal.numberOfPages	114 σ.	el
heal.fullTextAvailability	false