I have been tyring to understand the first condition given in the link <https://en.wikipedia.org/wiki/Regularity_structure> for quite some time now, at least a year. I have posted a similar question in <https://math.stackexchange.com/questions/3524946/how-to-think-of-a-set-that-has-no-accumulation-point> but did not get any reply. Please help me if possible.


As I understand the regularity structure deals with stochastic partial differential equations and this means we deal with stochastic variable or functions which are continous but not differentiable, this leads to discrete type of sets and hence we get the condition "no accumulation point" is this correct?


If so, the question is why it has to be bounded from below only? I was expecting it to be bounded from below as well as above or bounded.


Few words why this condition is necessary in regularity structure would be highly appreciated?


**Edit:** The comment from Nate Eldredge made it clear why it is not bounded from above, but I am still keeping this question just in case someone write in very consice terms the conection bteween the index set A and the regularity structure.



When and why do we require the condition that :"a subset bounded from below and has no accumulation points?"
From the references mentioned in the comments , (section 4.4,[*"Renormalisation of parabolic stochastic PDEs"*](https://arxiv.org/pdf/1803.03044.pdf)) (RP) and (section 6.1,*"Introduction to regularity structures"*) (IRS), we first start from the pde fixed point problem


$$\Phi = G \ast (\xi − \Phi^3 ) + G\Phi\_0 ,$$


where $\*$ denotes space-time convolution and where we denote by $G \Phi\_0$
the harmonic extension of $\Phi\_0$ and we split $G = K + \hat K$ into singular kernel $K$ and smooth $\hat{K}$. Then we consider the analogous *abstract* fixed point problem ((4.22) in (RP) and (6.3) in (IRS))


$$\Phi = \mathcal{K} \bigl(\Xi - \Phi^3\bigr) + \hat{\mathcal{K}} \bigl(\Xi - \Phi^3\bigr) + G \Phi\_0 $$


where you see that by repeated insertion of $\Phi$, you get combinations of *abstract symbols* eg. $\Xi, \mathcal{I}\Xi, \mathcal{I}(\Xi) \mathcal{I}( \mathcal{I}(\Xi)^3)$ and $\mathcal{I}$ is the "abstract integration" map against that kernel.


So the question becomes: how to keep track and organize all those new abstract expressions. As you see in that section, he gives specific examples of regularity structures based on generating sets of the above symbols.


What about the index set?


A big issue with SPDEs is the presence of the white noise $\xi$ because it is a distribution and so when it shows up in an SPDE, it can make them ill-posed. This is what creates the negative regularity degree. So as in the example, we incorporate that by assigning negative numbers to $\Xi$, (the abstract symbol for noise) eg. in the previous example $deg(\Xi)=-\frac{5}{2}-\epsilon$ for some $\epsilon>0$.


On the other hand, what saves us is integrating against the kernel $\mathcal{K}$ because as you might know from the heat equation, integrating against a heat kernel immediately smoothens functions. A similar phenomenon happens here via the Schauder estimates improving the regularity of the solution pushing it to the positive side. So we assign positive increase: $|\mathcal{I} \tau| = |\tau| + 2$


So there is this battle between lowering/increasing regularity. The index set
contains all the degree of the above symbols starting from the most negative degree of $\Xi$ eg. above $\{-\frac{5}{2}-\epsilon,-\frac{5}{2}-\epsilon+2, -\frac{5}{2}-\epsilon+4,...\} $


So the requirement "bounded below" ensures that eventually with enough convolutions, we will get actual functions out of this process rather than distributions (using reconstruction operator and removal of appropriate constants).


