Timeline for Inequalities about tripling and doubling sumsets
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Apr 3, 2020 at 16:05 | comment | added | George Shakan | You should check this paper of Gyarmati, Matolcsi and Ruzsa where they prove that $|3A| \geq 3/2|2A| - O(1)$. This was originally proved by Lev in core.ac.uk/download/pdf/82292093.pdf. It is weaker than what you want, but the proof is short | |
| Dec 8, 2018 at 16:29 | comment | added | Hailong Dao | @Seva: that's a good hint, I will look up his papers. I would still like to know if there are some criteria other than probabilistic ones (although they are probably the best one can say). | |
| Dec 8, 2018 at 16:20 | comment | added | Seva | There are lots of papers dealing with sumset inequalities, many of them due to Imre Ruzsa, but your particular inequalities hold almost always. | |
| Dec 8, 2018 at 16:15 | comment | added | Hailong Dao | @Seva: cool. So is there any hope for a non-trivial sufficient condition? Anything in the literature? | |
| Dec 8, 2018 at 15:59 | comment | added | Seva | No, it does not. Take $A$ to be a subset of $[0,l]$ such that $2A=[0,2l]$. Then $|3A|+|A|\le 3l+|A|+1<4l+2=2|2A|$ provided $|A|<l+1$. | |
| Dec 8, 2018 at 15:28 | comment | added | Hailong Dao | @Seva: I edited the question now. As you pointed out, the generic $A$ satisfies these inequalities. But, for example, does the inequality for Question 2 holds always when $d=1$? Thanks. | |
| Dec 8, 2018 at 15:22 | history | edited | Hailong Dao | CC BY-SA 4.0 |
added 104 characters in body
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| Dec 8, 2018 at 15:14 | comment | added | Hailong Dao | I am interested in finding concrete sufficient condition. Even for $d=1$. | |
| Dec 8, 2018 at 14:33 | comment | added | Seva | I do not understand then. For a set $A$ in a general position, one has $|3A|\sim |A|^3/6$, while $|2A|=O(|A|^2)$. Therefore, your inequalities hold true for any "typical" set, not necessarily an arithmetic progression or anything else of this sort? | |
| Dec 8, 2018 at 14:12 | history | edited | Hailong Dao | CC BY-SA 4.0 |
edited title
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| Dec 8, 2018 at 14:11 | comment | added | Hailong Dao | No, I am just not sure what is the standard notation. | |
| Dec 8, 2018 at 14:10 | comment | added | Seva | Is there any difference between your $s_i(A)$ and the quantity commonly denoted by $|iA|$ (the size of the $i$-fold sumset $iA$)? | |
| Dec 8, 2018 at 13:32 | history | asked | Hailong Dao | CC BY-SA 4.0 |