In mathematics, Roth's theorem or Thue–Siegel–Roth theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative...

Friedrich Roth FRS (29 October 1925 – 10 November 2015) was a German-born British mathematician who won the Fields Medal for proving Roth's theorem on the...

bounded by the square root of the modulus. Thue–Siegel–Roth theorem, also known as Roth's theorem, is a foundational result in diophantine approximation...

natural numbers. It was first proven by Klaus Roth in 1953. Roth's theorem is a special case of Szemerédi's theorem for the case k = 3 {\displaystyle k=3} ...

Szemerédi's theorem are trivial. The case k = 3, known as Roth's theorem, was established in 1953 by Klaus Roth via an adaptation of the Hardy–Littlewood circle...

part to its large number of consequences in number theory including Roth's theorem, the Mordell conjecture, the Fermat–Catalan conjecture, and Brocard's...

conjectures for which it gives a conditional proof. The consequences include: Roth's theorem on Diophantine approximation of algebraic numbers. The Mordell conjecture...

theorems in Diophantine geometry that are of fundamental importance include: Mordell–Weil theorem Roth's theorem Siegel's theorem Faltings's theorem Serge...

arithmetic progressions of length 3 is a consequence of an improved bound in Roth's theorem. A 2016 paper by Bloom proved that if A ⊂ { 1 , . . , N } {\displaystyle...

Siegel (1921), Freeman Dyson (1947), and Klaus Roth (1955), leading finally to the Thue–Siegel–Roth theorem: If x is an irrational algebraic number and ε...

Roth's theorem on 3-term arithmetic progressions, and a generalization of it, the hypergraph removal lemma, can be used to prove Szemerédi's theorem....

Thue–Siegel–Roth theorem says that, for algebraic irrational numbers, the exponent of 2 in the corollary to Dirichlet’s approximation theorem is the best...

where C is any real number satisfying C > 160/9. While the theorem is related to Roth's theorem, its real use lies in the fact that it is effective, in the...

theorem on the density of sets of integers that avoid longer arithmetic progressions. To distinguish Roth's bound on Salem–Spencer sets from Roth's theorem...

a\cdot S+h} . This theorem follows from the multidimensional corners theorem by a simple projection argument. In particular, Roth's theorem on arithmetic progressions...

Mordell conjecture. Together with Gisbert Wüstholz, he reproved Roth's theorem, for which Roth had been awarded the Fields medal in 1958. In 1994, he returned...

functions are linearly dependent. Roth used this result about generalized Wronskians in his proof of Roth's theorem. For more general conditions under...

Zeichenreihen", Selected Math. Papers of Axel Thue, Universiteitsforlaget: 67 Roth's theorem – Algebraic numbers are not near-rational Semi-Thue system – String...

the theorem unconditionally by combining a version of the Thue–Siegel–Roth theorem, from diophantine approximation, with the Mordell–Weil theorem from...

{\displaystyle q^{2}(\log q)^{1+\epsilon }} . Roth's work effectively ended the work started by Liouville, and his theorem allowed mathematicians to prove the transcendence...

all such pairs using Pell equations. It follows from the Thue–Siegel–Roth theorem that there are only a finite number of pairs of this type, but Størmer...

usually known as Liouville's theorem (on diophantine approximation), there being several results known as Liouville's theorem. Lemma: If α {\displaystyle...

known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, Siegel's method, Siegel's lemma and the...

|r|\leq Z} with Z → ∞ {\displaystyle Z\rightarrow \infty } ) Roth's theorem Faltings's Theorem A. Thue (1909). "Über Annäherungswerte algebraischer Zahlen"...

main examples were: The Thue–Siegel–Roth theorem Siegel's theorem on integral points, from 1929 The 1934 theorem of Hans Heilbronn and Edward Linfoot...

In mathematics, the subspace theorem says that points of small height in projective space lie in a finite number of hyperplanes. It is a result obtained...

1112/jlms/s1-28.1.104. MR 0051853. Zbl 0050.04002. Sanders, Tom (2011). "On Roth's theorem on progressions". Annals of Mathematics. 174 (1): 619–636. arXiv:1011...

Vojta. According to this analogy, 2 is the exponent in the Thue–Siegel–Roth theorem. On this analogy with number theory we refer to the survey of Lang (1987)...

theorem Gauss–Kuzmin–Wirsing operator Minkowski's question mark function Generalized continued fraction Kronecker's theorem Thue–Siegel–Roth theorem Prouhet–Thue–Morse...

F.; Sisask, Olof (2021-09-01). "Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions". arXiv:2007.03528 [math.NT]. Spalding, Katie...