By Thomas Bloom
In this blog post I will give my personal view on the recent counterexamples to the unit distance conjecture and sum-product conjecture over the reals (see [90] and [52] respectively). My goal is to sketch the constructions and try and give some intuition as to where they came from and why they work. My main target audience is the me-of-a-month-ago, who did not know much algebraic number theory, and who needs the relevant parts of the basic theory in this area explained, but wants to know exactly where the quantitative improvements come from. (I know a bit more algebraic number theory now, but still much less than I'd like!) My focus is on the combinatorial side, and I will stop with an appeal to the literature as soon as we need to do any serious number theory. (In particular I will not attempt to discuss even the statement, let alone the proof, of the Golod-Shafarevich theorem.)
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Thanks for the great and insightful blog post!
Thanks for the post, it is most helpful. I just wanted to add that the existence of towers of such number fields is an unexpected wonder. In his survey paper Odlyzko writes: "For a long time it was conjectured that if $d_n$ denotes the minimal root-discriminant of a number field of degree $n$, then $d_n \rightarrow \infty$ as $n \rightarrow \infty$. (This is known to be true for abelian fields.) If true, this would have shown that all Hilbert class field towers terminate, and so all number fields could be embedded in fields of class number one. However, Golod and Shafarevich showed that infinite Hilbert class field towers do exist. The best current results are due to Martinet who showed that there is an infinite ..."
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