Timeline for In what cases are the counting function and representation functions strongly related?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 16, 2011 at 17:28 | comment | added | user9072 | I believe doing the Fourier-analysis in a finite group is actually desirable, even better would be if one could do it in a vector-space over a finite field. I cannot sketch why this is so, but this is a reason why analogues of questions for the naturals are considered intensely in vector spaces over finite fields; some problem go away in this setting yet others persist, and one tries to make progress on the persisting ones, first, in the simplified setting. | |
| Feb 16, 2011 at 17:16 | comment | added | user9072 | Regarding linking natural numbers to a group: another strategy is to not consider all of $A$ but to take some (variable) cutoff $N$ and to consider $A \cap [1,N]$ as a subset of the cyclic group integers modulo $N$; or also integers modulo some $N'$ that depends on $N$, say $2N$ or a prime of size about $2N$ to get even a prime cyclic group. Then one investigates the problem in the finite cyclic group(s), and then 'lifts' the information to the naturals. For example various proofs of Roth-type results, proving the existence of arith. prog. under density assumptions, work like this. | |
| Feb 16, 2011 at 7:44 | comment | added | Thomas Bloom | Can't you just embed A into the integers and hence enjoy the ambient structure while only losing a factor of two or so? | |
| Feb 16, 2011 at 3:08 | comment | added | Stanley Yao Xiao | Right, linear bias is quite a strong indicator of whether $a(n)$ and $r_{A,h}(n)$ will be related or not. 'Random' or 'uniform' sets tend to enjoy having $a(n)$ and $r_{A,h}(n)$ be quite related. The problem is that a lot of the machinery for detecting linear bias, at least in Additive Combinatorics by Tao and Vu, rely on the ambient structure of a group, and the natural numbers of course are not a group. Hence I am looking for some other way to detect bias other than the Fourier-analytic ways in Tao-Vu. | |
| Feb 16, 2011 at 2:56 | history | edited | user9072 | CC BY-SA 2.5 |
added a qualification for clarity
|
| Feb 16, 2011 at 2:31 | history | edited | user9072 | CC BY-SA 2.5 |
add final remark
|
| Feb 16, 2011 at 2:22 | history | answered | user9072 | CC BY-SA 2.5 |