Timeline for How many rich directions does a set in $\mathbb F_p^2$ determine?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 6, 2018 at 19:20 | history | edited | Luca Ghidelli | CC BY-SA 4.0 |
added 3 characters in body
|
| Jun 6, 2018 at 14:40 | comment | added | Luca Ghidelli | I'd like to hear about that! Does your argument seem to extend to k-rich directions? I will try to see what is the precise bound one gets for |P|<2p. | |
| Jun 6, 2018 at 14:33 | comment | added | Seva | This is a great bound. I cannot recall now what application I had in mind (if any), but I certainly believe that this rich directions problem is interesting in its own rights. Interestingly, I have a somewhat incomplete argument (hopefully, it can be completed) which gives the same bound $p/3$ for sets of size $|P|=2p$ (but does not work for smaller sets). | |
| Jun 6, 2018 at 13:31 | history | edited | Luca Ghidelli | CC BY-SA 4.0 |
added 62 characters in body
|
| Jun 6, 2018 at 9:07 | comment | added | Luca Ghidelli | I worked out a lower bound linear in p. Does it help now? | |
| Jun 6, 2018 at 1:58 | history | edited | Luca Ghidelli | CC BY-SA 4.0 |
Before it was only a partial answer
|
| Jun 5, 2018 at 6:36 | comment | added | Seva | Oh, I see, thanks for the clarification; so, the result is essentially due to Redei, with some contribution by Megyesi, appeared first in Redei's book. BTW, as far as I understand, Szonyi's result cannot be derived from Redei-Megyesi using the latter as a "black box"? | |
| Jun 4, 2018 at 20:55 | comment | added | Luca Ghidelli | Sure, it's Rédei's book [1], Proposition 24'. Rédei had previously proved the upper bound (p+1)/2. The contribution of Megyesi is that he noticed that the value (p+1)/2 is actually impossible to attain. [1] Rédei, László. Polynome ¨uber endlichen K¨orpern, Birkh¨auser Verlag, Basel,1970. (English translation: Lacunary polynomial over finite fields, North Holland, Amsterdam, 1973.) | |
| Jun 4, 2018 at 20:45 | comment | added | Luca Ghidelli | Sure, it's Rédei's book L. R´edei. Polynome ¨uber endlichen K¨orpern, Birkh¨auser Verlag, Basel, 1970. (English translation: Lacunary polynomial over finite fields, North Holland, Amsterdam, 1973.) | |
| Jun 4, 2018 at 19:48 | comment | added | Seva | Do you have the reference to the original Rédei - Megyesi paper? | |
| May 2, 2018 at 13:01 | comment | added | Luca Ghidelli | The case |P|=p is already a theorem of Rédei and Megyesi (I think I should have included him as well). The theorem of Szonyi is the generalization to |P|<p. The argument I wrote can be used with |P'|<p, but I don't know what are the cases of equality in Szonyi's theorem. | |
| May 2, 2018 at 12:55 | comment | added | Luca Ghidelli | I see, I wonder if it is possible to have Rédei-type results for "high blocking sets" to have a lower bound on high directions. | |
| May 2, 2018 at 5:43 | comment | added | Seva | And, BTW, why do you say "in the theorem of Redei"? Isn't it a theorem of Szonyi? | |
| May 2, 2018 at 5:41 | comment | added | Seva | Thanks, this is interesting, but I actually need a lower bound: what is the number of rich directions that any set $P$ is guaranteed to determine? | |
| May 2, 2018 at 5:15 | history | answered | Luca Ghidelli | CC BY-SA 4.0 |