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pro vyhledávání: '"Bloom, Thomas F."'
In this expository note, we give a concise and accessible introduction to the real-analytic determinant method for counting integral points on algebraic curves, based on the classic 1989 paper of Bombieri-Pila.
Comment: 17 pages. arXiv admin not
Comment: 17 pages. arXiv admin not
Externí odkaz:
http://arxiv.org/abs/2312.12890
Autor:
Bloom, Thomas F., Kuperberg, Vivian
We prove near-optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application we prove near-optimal upper bounds for the average of the refined
Externí odkaz:
http://arxiv.org/abs/2312.09021
Autor:
Bloom, Thomas F., Sisask, Olof
In a recent breakthrough Kelley and Meka proved a quasipolynomial upper bound for the density of sets of integers without non-trivial three-term arithmetic progressions. We present a simple modification to their method that strengthens their conclusi
Externí odkaz:
http://arxiv.org/abs/2309.02353
Autor:
Bloom, Thomas F.
We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p\textrm{ prime}\}$. The same conclusion holds for the set $\{p-h : p\textrm{ prime}\}$ for any $h\in\mathbb{Z}\backslash\{0\}$, provided a
Externí odkaz:
http://arxiv.org/abs/2305.02689
Autor:
Bloom, Thomas F., Sisask, Olof
Publikováno v:
Ess. Number Th. 2 (2023) 15-44
We give a self-contained exposition of the recent remarkable result of Kelley and Meka: if $A\subseteq \{1,\ldots,N\}$ has no non-trivial three-term arithmetic progressions then $\lvert A\rvert \leq \exp(-c(\log N)^{1/11})N$ for some constant $c>0$.
Externí odkaz:
http://arxiv.org/abs/2302.07211
Autor:
Bloom, Thomas F., Elsholtz, Christian
Any rational number can be written as the sum of distinct unit fractions. In this survey paper we review some of the many interesting questions concerning such 'Egyptian fraction' decompositions, and recent progress concerning them.
Comment: 10
Comment: 10
Externí odkaz:
http://arxiv.org/abs/2210.04496
Autor:
Bloom, Thomas F.
We prove that any set $A\subset \mathbb{N}$ of positive upper density contains a finite $S\subset A$ such that $\sum_{n\in S}\frac{1}{n}=1$, answering a question of Erd\H{o}s and Graham.
Comment: 18 pages, appendix co-written by Bhavik Mehta
Comment: 18 pages, appendix co-written by Bhavik Mehta
Externí odkaz:
http://arxiv.org/abs/2112.03726
Autor:
Bloom, Thomas F., Maynard, James
We show that if $A\subset \{1,\ldots,N\}$ has no solutions to $a-b=n^2$ with $a,b\in A$ and $n\geq 1$ then \[|A|\ll \frac{N}{(\log N)^{c\log\log \log N}}\] for some absolute constant $c>0$. This improves upon a result of Pintz-Steiger-Szemer\'edi.
Externí odkaz:
http://arxiv.org/abs/2011.13266
Autor:
Bloom, Thomas F.
(This text is a survey written for the Bourbaki seminar on the work of F. Manners.) Gowers uniformity norms are the central objects of higher order Fourier analysis, one of the cornerstones of additive combinatorics, and play an important role in bot
Externí odkaz:
http://arxiv.org/abs/2009.01774
Autor:
Bloom, Thomas F., Sisask, Olof
We show that if $A\subset \{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions then $\lvert A\rvert \ll N/(\log N)^{1+c}$ for some absolute constant $c>0$. In particular, this proves the first non-trivial case of a conjecture of
Externí odkaz:
http://arxiv.org/abs/2007.03528