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Nov
19 |
awarded | Nice Question |
Nov
18 |
accepted | Where did the term “additive energy” originate? |
Nov
18 |
asked | Where did the term “additive energy” originate? |
Nov
17 |
comment |
Is there a “complete” Sidon sequence?
A precise formulation of the result Ben references is as follows: if the representation function of a set $A$ satisfies $r(n) \leq 1$ for all $n$ then $|A \cap [1,...,n]| \leq C \sqrt{n/ \log n}$ holds infinitely often. It is open if $r(n) \leq 2$ implies $|A \cap [1,...,n]| \leq o( \sqrt{n} ) $. The problem of showing $r(n) \leq k$ implies $|A \cap [1,...,n]| \leq o( \sqrt{n} ) $ for all $k$ implies a famous longstanding problem of Erdos and Turan. See: mathoverflow.net/questions/43995/… |
Nov
8 |
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What are some very important papers published in non-top journals?
There are other examples among Ruzsa's papers that I am surprised have not yet been mentioned. One example (the OP is familiar with) is Ruzsa's approach to constructing Sidon sequences that overcame a fundamental obstacle on a longstanding open problem: "I. Ruzsa, An infinite Sidon sequence, J. Number Theory 68 (1998)." Another is his Fourier analytic approach to Freiman's theorem which has been very influential in arithmetic combinatorics: "I. Ruzsa, Generalized arithmetical progressions and sumsets". Acta Mathematica Hungarica 65 (4): 379–388." |
Nov
8 |
awarded | Nice Answer |
Nov
7 |
answered | What are some very important papers published in non-top journals? |
Nov
4 |
comment |
Partition regularity in the squares
See "Uniformity of multiplicative functions and partition regularity of some quadratic equations" by Frantzikinakis and Host at arxiv.org/abs/1303.4329 for some related discussion. |
Oct
29 |
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Subsets of [1..N] with no three-term arithmetic progressions and no large gaps
If Behrend's set had large gaps (that is a gap much larger than the typical gap suggested by the density) then one of the "sides" of the set excluding the gap would have increased density and still not contain three-term AP's. It seems that one should be able to iterate this observation to obtain either (a) a set more dense than Behrend without 3-term APs, or (b) a set of similar density to Behrend where the maximal gap is near the expected size. |
Oct
21 |
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Additive energy of Piatetski-Shapiro sequences
I believe what you say is true however this really isn't an answer unless you can fill in the details or provide a reference. We know things must be somewhat subtle given that such statements aren't known for the full range of $c$. |
Oct
20 |
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Additive energy of Piatetski-Shapiro sequences
Using Elekes' relation between sum-product and incidence theorems (see the sketch in my answer here mathoverflow.net/questions/217557/…), this should follow from a Szemeredi-Trotter theorem for translates of the curve $(x, \lfloor x^{c} \rfloor )$. However the crossing number proof of Szemeredi-Trotter only uses that translates of this curve intersects in $O(1)$ places and should apply here. |
Oct
20 |
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Additive energy of Piatetski-Shapiro sequences
I haven't checked this carefully but here is an alternate approach that seems simpler. By BSG / "routine" additive combinatorics it suffices to show that $|A+A| > |A|^{1+c}$ for some fixed $c>0$ and $A$ a large subset of $[N]$ (that is $|A|>N^{1-a}$ for some small $a$). It thus should suffice to show that $\max(|A+A|, |f(A)+f(A)|) > |A|^{1+c}$ for $c>0$ where $f(x) = \lfloor x^{c} \rfloor$. |
Oct
20 |
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Additive energy of Piatetski-Shapiro sequences
... See the introduction of this paper "Exceptional set of a representation with fractional powers" by Balanzario, Garaev, and Zuazua for a discussion of this result. This isn't quite a proof since one needs to consider all restricted sumsets of positive density within $A + A$ and not just the full set, however one might be able to adapt the ideas from that proof or use the "restriction theorem" for the P-S sequence of Mirek to pass to the general case (arxiv.org/abs/1305.0043). |
Oct
20 |
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Additive energy of Piatetski-Shapiro sequences
I believe one should be able to prove this with existing technology. Let me give an incomplete sketch (which is in line with the ideas of previous comments). First by the Balog-Szemeredi-Gowers lemma if the claim is false then one would have that every "restricted" sumset of positive density in $A \times A$ has small doubling. That is $|A + A| \lesssim N^{1/c+o(1)}$ (where the sumset is restrict to a positive proportion of the set $A \times A$). On the other hand it is known for $c<3/2$ the sumset of the Piatetski-Shapiro sequence has positive density in the integers. ... |
Oct
18 |
awarded | Yearling |
Oct
7 |
comment |
Brownian motion, quadratic variation, existence of partitions?
It still seems a bit unclear but it's probably just me. In any event I have reworded my answer. |
Oct
7 |
revised |
Brownian motion, quadratic variation, existence of partitions?
added 55 characters in body |
Oct
7 |
comment |
Brownian motion, quadratic variation, existence of partitions?
I should point out the difference in our answers depends on one's interpretation of the question. In exercise 1.13(a) the sequence of partitions are random (that is they vary with each element of the probability space). In this case Taylor's result gives the exact asymptotic dependence. If one uses a deterministic sequence of partitions, which is how I read the question, the answer is 'no' based on Lévy's work. |
Oct
7 |
answered | Brownian motion, quadratic variation, existence of partitions? |
Oct
7 |
asked | Is there a decomposition strengthening of the Sauer-Shelah Lemma? |