User kiskis - MathOverflow

Answer by kiskis for Examples of conjectures that were widely believed to be true but later proved false

2012-07-03T01:06:25Z

Luzin's conjecture was widely believed to be false, until it was proven by Carleson in 1966.

I'm citing from Lennart Carleson's biography: "In 1913 Luzin conjectured that if a function $f$ is square integrable then the Fourier series of $f$ converges pointwise to $f$ Lebesgue almost everywhere. Kolmogorov proved results in 1928 which seemed to suggest that Luzin's conjecture must be false but Carleson amazed the world of mathematics when he proved Luzin's long-standing conjecture in 1966. He explained how he was led to prove the theorem:-

... the problem of course presents itself already when you are a student and I was thinking about the problem on and off, but the situation was more interesting than that. The great authority in those days was Zygmund and he was completely convinced that what one should produce was not a proof but a counter-example. When I was a young student in the United States, I met Zygmund and I had an idea how to produce some very complicated functions for a counter-example and Zygmund encouraged me very much to do so. I was thinking about it for about 15 years on and off, on how to make these counter-examples work and the interesting thing that happened was that I realised why there should be a counter-example and how you should produce it. I thought I really understood what was the background and then to my amazement I could prove that this "correct" counter-example couldn't exist and I suddenly realised that what you should try to do was the opposite, you should try to prove what was not fashionable, namely to prove convergence. The most important aspect in solving a mathematical problem is the conviction of what is the true result. Then it took 2 or 3 years using the techniques that had been developed during the past 20 years or so"

How small can {log p/q} be?

2012-06-30T21:01:20Z

Denote by $||x||$ the distance between $x$ and the nearest integer. Mahler conjectured that there is a constant $c > 0$ such that for any integer $n \geq 2$ $$ ||\log n|| \geq n^{-c} $$ and Waldschmidt made the stronger conjecture that $$ ||\log n|| \geq (\log n)^{-c} $$ I am interested in the analogous question for rational numbers rather than integers. Namely, given two integers $0 < p \neq q \leq H$ are there any results and/or conjectures regarding how small $$ ||\log (p / q)|| $$ can be in terms of $H$ ? A lower bound of the form $H^{-1 - \varepsilon}$, for some small $\varepsilon > 0$ would be ideal but I'm not sure how much one can hope for. (Notice that $H^{-1}$ is attained when $q = H$ and $p = q-1$, indeed in that case by a Taylor expansion $\log (p/q) = \log(1 - 1/H) \asymp H^{-1}$)

Comment by kiskis

2012-06-30T22:18:58Z

As Mahler's conjecture suggest these could be much stronger than what follows from Baker

Comment by kiskis

2012-06-30T22:18:26Z

I also would like to know the conjectures regarding the "truth"