can somebody please answer this question for me I am having a lot of trouble proving this. I need proof

Let $M$ a nonparabolic Riemannian manifold. If exists only one nonparabolic end $E$. We would like to know why the subspace of space of bounded harmonic functions with finite Dirichlet integral is the space of constant functions. Well, by the existence of nonparabolic end $E$, i can build a harmonic $g$ function in $M- \Omega$( where $\Omega$ is subdomain compact of $M$ such that $E$ is a nonparabolic end of $M$). The function $g$ is limited and has finite Dirichlet integral. But i don't know as to use the fact which has only one nonparabolic end of $M$.

Let $\mathcal O$ be an order in an imaginary quadratic field $K$.

Does there exists an element $\lambda\in \mathcal O$ such that the norm $N(\lambda)$ is not a square?

Does there exists an element $\lambda\in \mathcal O$ such that the norm $N(\lambda)$ is squarefree and not equal to $1$?

**Is there an elementary solution for 1. and 2.?**

Note that if there was a simple proof of the first statement, then we could perhaps simplify the proof of the integrality of the $j$-invariant at $CM$ points by avoiding reduction to the case of the maximal order.

F-polynomials are certain polynomials appears in the expansion formula of a cluster variable, see for example the formula (6.5) in cluster algebras IV. Theta functions in the paper correspond to cluster variables. Are there some relations between F-polynomials and theta functions? Thank you very much.

I have been struggling with some linearization argument of the following paper: "M. Weinstein: Modulational stability of ground states of NLS". In order to give a bit of context to my question, let us consider the NLS equation $$ 2i\phi_t+\Delta\phi+\vert\phi\vert^{2\sigma}\phi=0, \quad 0<\sigma<\tfrac{2}{n-2}.$$ This equation has very interesting "localized" solutions of the form: $$\phi(t,x)=u(x)e^{it/2},$$ where $u(t,x)$ solves $\Delta u-u+\vert u\vert^{2\sigma}u=0$. Besides, the latter equation has an even more interesting real, positive and radial $H^1(\mathbb{R}^n)$ solution called "Ground state" and denoted by $R(x)$.

Now let me try to explain my question. Consider the perturbed Initial Valued Problem (IVP): $$2i\phi_t^\varepsilon+\Delta \phi^\varepsilon+\vert \phi^\varepsilon\vert^{2\sigma}\phi^\varepsilon=\varepsilon F(\vert \phi^\varepsilon\vert)\phi^\varepsilon, \quad \phi^\varepsilon(t=0,x)=R(x)+\varepsilon S(x)$$ We will seek solutions of the previous equation of the form $$\phi^\varepsilon(t,x)=(R(x)+\varepsilon w_1+\varepsilon^2 w_2+...)e^{it/2}.$$ According to Weinstein if you reeplace this function into the perturbed equation and linearize you will get the following IVP for the linearized perturbation $w$: $$ 2iw_t+\Delta w-w+(\sigma+1)R^{2\sigma}w+\sigma R^{2\sigma}\overline{w}=F(R)R, \quad w(0,x)=0.$$ Now my problem is: I do not really understand how to obtain this linearization, can someone explain a little bit how to do it? Or recommend some references to learn about it. I tried replacing $\phi^\varepsilon$ (truncated after $\varepsilon w$) and then I took the derivative with respect to $\varepsilon$ and evaluate at $\varepsilon=0$, but I cannot recover the equation claimed by Weinstein, so I think that I'm not undersitanding how to linearize.

Note2: The parameter $n$ denotes the dimension.

A string over a finite alphabet $A$ can be thought of as a function $f:\{1,2,...,m\} \rightarrow A$ for some natural number $m$.

A 2-Dim string over $A$ is a function $f$ where $f:\{1,2,\ldots,m\}\times\{1,2,\ldots,n\} \rightarrow A$ for some natural numbers $m$ and $n$.

If $f$ and $f'$ are 2-Dims (over $A$), with dimensions $(m,n)$ and $(m'n')$ respectively, we say $f\leq f'$ if there is are increasing functions $D:\{1,2,\ldots,m\}\rightarrow\{1,2,\ldots, m'\}$ and $E:\{1,2,\ldots,n\}\rightarrow\{1,2,\ldots, n'\}$ such that whenever $1\leq i\leq m$ and $1\leq j\leq n$, $f(i,j)=f'(D(i),E(j))$.

Is there a proof or disproof of the version of Higman's lemma that states that for every infinite sequence $f_1,f_2,\ldots$ of 2-Dims over $A$, there is $1\leq u<v$ such that $f_u\leq f_v$?

recently I was reading the article ''On the Ramsey property for sets of reals'' of Ilias Kastanas (https://www.jstor.org/stable/2273667?seq=1#metadata_info_tab_contents), in this article the author characterizes the sets completely Ramsey by a game created by Kastanas, the proof seemed very good. But I think it can be formalized better, for example in the proof of that Player 1 has a winning strategy in the Kastanas game then there is a homogeneous set in $\psi$, to build that set the author use a tree, but I think that the proof is more intuitive than formal, I would like to know if someone knows any more formal proof, or more bibliography about it.

Thank you

I need to know, or at least have a good bound for, the asymptotic behaviour on $x$ of amount of integers less o equal than $x$ that are square free and with exactly $k$ primes on its decomposition. That is the cardinal of the following set $$ \mathcal{J}_T(x,k) = \{ n \in \mathbb{N} : n \le x, \Omega(n)=k , n \mbox{ is square free } \}.$$ Other way to describe this cardinal, using the Möbius function, is $$ |\mathcal{J}_T(x,k) | = \sum_{\Omega(n) = k, n \le x } |\mu(n)|.$$

I am looking for the asymptotic behaviour on $x$, but this will depend also on $k$ in some way. The bound given using
$$
\sum_{\Omega(n) = k, n \le x } |\mu(n)| \le \sum_{n \le x } |\mu(n)| \le \frac{6x}{\pi^2} + O(\sqrt{x}) \ll x,
$$

is not good enought for my purposes.

Thanks in advanced, any reference or idea is helpful

Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair.

Consider the complexified tangent space $TM \otimes \mathbb{C}$ and the complex-linear extension of $R$, which we still denote $R$. By definition, $N$ has *nonpositive Hermitian sectional curvature* if $R(X, Y, \bar{X}, \bar{Y}) \leqslant 0$ for all $X, Y \in TM \otimes \mathbb{C}$.

Obviously, nonpositive Hermitian sectional curvature is stronger than nonpositive sectional curvature.

**QUESTION.** Is nonpositive Hermitian curvature *strictly* stronger than nonpositive curvature?

In other words, are there examples of Riemannian manifolds with nonpositive sectional curvature, but not nonpositive Hermitian sectional curvature?

I expect the answer easily yes, in fact it is claimed in e.g. [1] or [8], but I couldn't find an example in the relevant literature, e.g. [1][2][3][4][5][6][7][8][9].

NB: Yau-Zheng [8] showed that the answer is no for manifolds with negative $\delta$-pinched sectional curvature with $\delta \geqslant 1/4$. According to [9, Theorem 9.26], the answer is no for Kähler surfaces.

$$$$

**FOLLOW UP QUESTIONS**

Following (almost) the terminology of Siu [6], a Riemannian manifold with nonpositive Hermitian sectional curvature has *strongly nonpositive curvature*. He also introduces *very strongly nonpositive curvature*: Consider the curvature operator
$$
\begin{aligned}
Q \colon \otimes^2 TM \times \otimes^2 TM \to \mathbb{R}
\end{aligned}
$$
such that $Q$ is defined for decomposable tensors by $Q(X\otimes Y, Z \otimes W) = R(X , Y, Z , W)$. $N$ has *very strongly nonpositive curvature* if $Q(\sigma, \sigma) \leqslant 0$ for all $\sigma \in \otimes^2 TM$. In other words, the curvature operator is negative semidefinite.
In this case, the complex-linear extension of $Q$ is still negative semidefinite, which clearly implies that $M$ has strongly nonpositive curvature.

**Question 2.** Is there an example showing that very strongly nonpositive curvature is strictly stronger than strongly nonpositive curvature?

Finally, there is a notion of (very) strongly negative curvature for Kähler manifolds, but it's not simply something like $Q(\sigma, \sigma) < 0$ for all nonzero $\sigma$. Indeed, still denoting $Q$ its complex-linear extension, we have $Q(\sigma, \bar{\sigma}) = 0$ for any $\sigma$ of type $(2,0)$ or $(0,2)$, e.g. $X \otimes Y$ with $X, Y \in T^{1,0} M$. $N$ has *very strongly negative curvature* if $Q(\sigma, \bar{\sigma}) < 0$ for all nonzero tensors $\sigma \in T^{1,0} M \otimes T^{0,1} M$, and $N$ has *strongly negative curvature* if $Q(\sigma, \bar{\sigma}) < 0$ for all nonzero tensors $\sigma \in T^{1,0} M \otimes T^{0,1} M$ of length $\leqslant 2$, e.g. $\sigma = X \otimes \bar{Y} + Z \otimes {\bar{W}}$.

It is clear that $$\text{very strongly negative} ~\Rightarrow~ \text{strongly negative} ~\Rightarrow~ \text{negative sectional curvature}$$

**Question 3.** Are there examples proving that the converse implications are false?

Again, according to [9, Theorem 9.26], the answer is no for Kähler surfaces.

Remark: Of course, there are similar notions of (very) strong nonnegative / positive curvature and one could ask the same questions.

$$$$

[1] J. Amorós, M. Burger, K. Corlette, D. Kotschick, and D. Toledo. Fundamental groups of compact Kähler manifolds. 1996.

[2] Eells and Lemaire. Two reports on harmonic maps. 1995

[3] Jost and Yau. Harmonic mappings and Kähler manifolds. 1983.

[4] Mostow and Siu. A compact Kähler surface of negative curvature not coveredby the ball. 1980.

[5] Ohnita and Udagawa. Stability, complex-analyticity and constancy of pluriharmonic maps from compact Kaehler manifolds. 1990.

[6] Siu. The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. 1980.

[7] Xin. Geometry of harmonic maps. 1996

[8] Yau and Zheng. Negatively $\frac14$-pinched Riemannian metric on a compact Kähler manifold.

[9] F. Zheng, Complex differential geometry, 2000.

I'm looking for a reference which gives sufficient conditions for the variance to be positive in the central limit theorem for Markov chains (cf https://en.wikipedia.org/wiki/Markov_chain_central_limit_theorem). In other words, using the notation on the wikipedia page, I'm looking for a sufficient condition for $\sqrt{n}(\hat\mu_n-\mu)$ to legitimately be a random variable in the limit.

In my situation, the Markov chain is simply a random walk on a finite, aperiodic, strongly-connected graph, so a result relating to this simpler situation would be good enough if nothing more general exists. Also, in my situation, actually attempting to estimate $\sigma^2$ via the formula is pretty much out of the question.

I beleive that I can argue that the variance will be positive if there are two directed cycles $v_0\to v_2\to\dots\to v_n=v_0$ and $u_0\to u_2\to\dots\to u_m=u_0$ with ${1\over n}\sum_{t=1}^n g(v_t)\neq{1\over m}\sum_{t=1}^m g(u_t)$. Intuitively, this is because the chain will always have a chance to move around either of these cycles and these cycles contribute different values to the sum.

I'm not looking for anyone to write down a proof since, if necessary, I can likely prove it myself. However, I would really like to avoid having to spend the necessary pages to do so in a paper I'm writing. It seems very likely that such a result must have appeared in either a paper or book somewhere, but I haven't had any luck in hunting down a reference so far.

It is well-known that the category of discrete fibrations over a category $\mathbb{C}$ is equivalent to the category of presheaves on $\mathbb{C}$.

More generally I think it is true, and probably well-known, though I can't find a reference, that the following two categories are equivalent:

- the category with objects: discrete fibrations $\mathbb{E} \to \mathbb{B}$ and morphisms: pairs of functors ($\mathbb{E}\to \mathbb{E}'$, $\mathbb{B} \to \mathbb{B}'$) making the square commute
- the category with objects: pairs ($\mathbb{B}$, $P_\mathbb{B}: \mathbb{B}^\mathrm{op} \to \mathbf{Set}$) consisting of a category and a presheaf on it, and morphisms: pairs $(f : \mathbb{B} \to \mathbb{B}', \alpha : P_\mathbb{B} \Rightarrow P_{\mathbb{B}'} \circ f^\mathrm{op})$ consisting of a functor and a natural transformation.

Suppose $\mathbb{B}$ is a site (i.e. has a topology), then the full subcategory of separated presheaves on $\mathbb{B}$ is a reflective subcategory of the category of all presheaves.

My question is:
Consider the variation of (2) above where the objects are ($\mathbb{B}, P_\mathbb{B}$) where $\mathbb{B}$ is a site. Call this $\mathbf{C}$.

Is it true that the full subcategory of $\mathbf{C}$ whose objects are ($\mathbb{B}, P_\mathbb{B}$) with $P_\mathbb{B}$ separated is a reflective subcategory of $\mathbf{C}$?

I have learned that an orientation of a manifold $M$ with respect to ordinary cohomology is an ordinary orientation, that an orientation with respect to complex K-theory is a Spin$^c$ structure, and that an orientation with respect to real K-theory is a spin structure. I think this is a very beautiful picture and I am wondering if orientations with respect to other theories like elliptic cohomology, G-equivariant cohomology, quaternionic K-theory, or spin cobordism correspond to interesting and well-studied differential-geometric structures.

Complex manifolds should be oriented with respect to any complex-oriented cohomology theory. Indeed, if $E$ is a complex-oriented cohomology theory then all complex vector bundles carry $E$-orientations. In particular, if $X$ is a complex manifold then its tangent bundle $TX$ has a complex structure making $X$ an $E$-oriented manifold.

Given that complex cobordism is the universal complex-oriented cohomology theory, I would guess that an orientation with respect to complex cobordism is a complex structure. I have been unable to find any literature on this and I am unsure how to approach the problem rigorously on my own. Maybe someone knows?

I would like to find a proof for Remark 0.5 in the following article of Claire Voisin:

https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/fanosymp.pdf

She writes in this remark the following:

**Remark 0.5** *A compact Kahler manifold $X$ which is rationally connected satisfies
$H^2(X, {\cal O}_X) = 0$, hence is projective.*

I understand that a Kahler manifold with $H^2(X, {\cal O}_X) = 0$ is projective. However, I don't understand why a Kahler manifold that is rationally connected has $H^2(X, {\cal O}_X) = 0$. Indeed, the definition for rational connectedness that Voisin is using is the following:

**Definition 0.3** *A compact Kahler manifold $X$ is rationally connected if for any two points $x, y\in X$, there exists a (maybe singular) rational curve $C\subset X$ with the property that $x\in C$, $y\in C$.*

So my question is the following: How to prove that $H^2(X, {\cal O}_X)$ for a compact Kahler manifold $X$ that satisfies Definition 0.3? Is this easy/hard/well-known?

**PS.** As Donu Arapura correctly says below the vanishing of $H^2(X, {\cal O}_X)$ for rationally connected projective manifolds is a classical fact. However I want a proof of such a vanishing for Kahler manifolds (to show that they are protective). So I want to know if this vanishing is a well known fact or a couple of pages are needed to prove it?

It is known that on an infinite board, if all squares of the form $(ki,kj)$ are removed, $k$ even, $i,j\in\mathbf{Z}$, then there is no knight's tour due to unbalanced black and white squares.

My questions are the following:

If $k$ is odd, does there exist a knight's tour? For example, $k=3$.

If finitely many black squares are to be removed, can there possibly exist a knight's tour?

Any results on these problems? Thanks.

Suppose $f_0,f_1, \ldots$ is a martingale (or i.i.d sequence) in $\mathbb R^d$ with $f_0=0$ and all $\|f_n - f_{n-1}\| \le L$ say. There are many concentration results for the initial segment of the martingale. For example Theorem 3.5 of this paper of Pinelis leads to the following variant of the Azuma-Hoeffding inequality.

$$P(\|f_n\| \ge \epsilon \text{ for some }n\le N) \le \exp\left (-\frac{\epsilon^2}{2NL^2} \right).\tag{1}$$

In the paper such results are called tail inequalities for martingales. What I'm interested in could also be called a tail inequality, except I am interested in is the behaviour after $N$ rather than before. Of course there's no reason to believe since $f_1,f_2,\ldots$ should remain bounded, but if we instead focus on the normalised values $\frac{f_n}{n}$ we can get some bounds. For example taking $\epsilon = N$ the Pinelis theorem implies

$$P\bigg (\frac{1}{N}\|f_N\| \ge 1\bigg) \le \exp\left (-\frac{N}{2L^2} \right).$$

The crudest thing we can do is take a union bound to get

$$P\bigg (\frac{1}{n}\|f_n\| \ge 1\text{ for all }n\ge N\bigg) \le \sum_{n=N}^\infty\exp\left (-\frac{n}{2L^2} \right).$$

The sum can be bounded by the integral

$$\sum_{n=N}^\infty\exp\left (-\frac{n}{2L^2} \right) \le \int_{N-1}^\infty\exp\left (-\frac{x}{2L^2} \right) = 2L \exp\left (-\frac{N-1}{2L^2} \right)$$

which goes to zero as $N \to \infty$.

I am wondering if there are any more sophisticated approaches to get better bounds?

One idea I had was, instead of forcing each $\frac{1}{n}\|f_n\| <1$ we force $\frac{1}{N}\|f_N\|_2 <1/2$. This forces the next $\frac{1}{n}\|f_n\| <1$ for all $n \le \left(\frac{2L}{2L-1/2}\right)N$. Then for $n_1 = \left(\frac{2L}{2L-1/2}\right)N$ we force $\frac{1}{n_1}\|f_{n_1}\|_2 <1/2$ and so on. Proceeding like this we get a union bound over $N,n_1,n_2,\ldots$ leading to a series of the form

$$\sum_{n=1}^\infty \exp \left( \frac{N}{8L^2} \left(\frac{2L}{2L-1/2}\right)^{n-1}\right).$$

and the integral

$$\int_{0}^\infty \exp \left( a b^{x-1}\right) = \int_{-1}^\infty \exp \left( a b^{x}\right)$$

for the obvious constants. Under the substitution $u =ab^{x}$ this becomes the exponential integral function

$$\frac{1}{\log b}\int_{a/b}^\infty \frac{e^{-t}}{t}dt = \frac{\text{Ei}_1(a/b)}{\log b}.$$

Using some special function inequalities I can bound the above by

$$\frac{e^{-a/b}}{(a/b)\log b}$$

which simplifies to something of the form

$$\frac{C \exp \left( \frac{N}{8L^2} \frac{2L-1/2}{2L}\right) }{N}$$.

We have acquired a $N$ in the denominator, and maybe a smaller coefficient $C$ than before. Unfortunately this makes no difference asymptotically because the coefficient inside the exponential is smaller than before.

You can also replace $1/2$ with any $\delta \in (0,1)$, perform the calculations, and try to minimise the result with respect to $\delta$. There is a closed form solution for such a $\delta$ but it is the solution of a cubic equation so doesn't offer much insight.

One could also try different $\delta_i$-values between each $n_i$ and $n_{i+1}$ but I cannot see how to bound the resulting series with an integral.

Has this problem been considered before? Could anyone provide a reference?

Let (for concreteness) $a = 2$, $b = \sqrt{5}$ and $\varphi = (\sqrt{5}+1)/2$. I am interested in solutions $(w,x,y,z) \in \mathbb{Z}[\varphi]^4$ of the system

$$ w^2 - ax^2 -by^2 + abz^2 = 1 $$ $$ \lvert w^2 + ax^2 +by^2 + abz^2 \rvert \ll \infty $$ $$ \lvert \bar{w}^2 + a\bar{x}^2 -b\bar{y}^2 - ab\bar{z}^2 \rvert \le C $$ for some constant $C$. Here $\overline{\alpha + \beta\sqrt{5}} = \alpha - \beta\sqrt{5}$ and ``$\ll \infty$'' means that ideally I would like to enumerate solutions in increasing order of this value. (Restriction of scalars turns this problem into a system of two quadratic equations and two inequalities in eight variables in $\mathbb{Z}$; if someone wants to see it, I can write it out including potential mistakes).

- What is the best (or even any practical) way to produce these?

I am aware that there is a lot of classical mathematics associated to this question but I don't quite manage to put it together. Perhaps a subquestion is:

- Can one enumerate the squares $s$ in $\mathbb{Z}[\varphi]$ with $\lvert \bar{s} \rvert \le C$ in increasing order of $\lvert s \rvert$?

Context: Let $k = \mathbb{Q}(\varphi)$ and let $A$ be the quaternion algebra $(\frac{a,b}{k})$ with norm $\nu$. With the above values the algebra $A$ is a skew field but tensoring with $\mathbb{R}$ in the two possible ways (taking $\sqrt{5}$ to $\pm\sqrt{5}$) gives an isomorphism with $M_2(\mathbb{R})$ which we equip with the map $$ \left\lVert\left(\begin{array}{cc}\alpha&\beta\\\gamma&\delta\end{array}\right)\right\rVert = \frac{1}{2}\left(\alpha^2 + \beta^2 + \gamma^2 + \delta^2\right). $$ The above system then asks for solutions $\lambda$ in the maximal order of $A$ for $\nu(\lambda) = 1$, $\lVert\lambda\rVert_{\sqrt{5} \mapsto \sqrt{5}} \ll \infty$ and $\lVert \lambda \rVert_{\sqrt{5} \mapsto -\sqrt{5}} \le C$.

The following is essentially a partial case for my previous question.

Let $B\subset\mathbb{R}^m$ be the unit ball with respect to a concrete norm on $\mathbb{R}^m$, say $l^p$-norm, $p\in (1,\infty)$. Let $v_1,...,v_n\in \mathbb{R}^m$ be linearly independent.

How to calculate the $n$-dimensional volume of $B\cap span\{v_1,...,v_n\}$?

I need to express this volume through the coordinates of $v_1,...,v_n$, or perhaps through some distances between certain combinations of them. I know that there is extensive literature on related matters, but I hope that this specific question has a specific answer..

My question mainly concern about how to construct a elliptic equation with Neumann boundary condition from a minimization problem.

Let $B=B_1 \subset \mathbb{R}^3$ and $E : H^1(B) \to \mathbb{R}$ $$E(u)= \int_{B}|\nabla u|^2+(u^2-1)^2 dx - \int_{\partial B}Q(u)d\mathcal{H}^2$$ We assume that $u_0 \in W^{1,2}$ to be the minimizer of the functional $E$ in the configuration space $$K=\{u\in W^{1,2}(B:\mathbb{R})\}.$$ Since $u_0$ is the critical point of the functional, we let $\xi \in K$, we obtain the equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \xi dx - \int_{\partial B }Q'(u)\xi d\mathcal{H}^2 = 0. $$ If we further require that $\xi$ vanishes on the boundary, we have the EL equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \,\xi dx = 0. $$ Suppose we also have that $u \in H^2(B)$, we have $$\int_B -\Delta u \, \xi + 4(u^2-1)u \xi dx + \int_{\partial B } \dfrac{\partial u}{\partial n}\xi- Q'(u)\xi d\mathcal{H}^2 = 0. $$ We finally obtain the equation $$\Delta u = 4(u^2-1)u \,\text{ in } B \,\text{ and }\, \dfrac{\partial u}{\partial n}=Q'(u) \,\text{ on }\, \partial B.$$

My main goal is to prove the minimizer $u_0$ solve the above equation weakly with the desried Neumann boundary condition. However, my question is how to obtain the $H^2$ bound of $u$? I think we can apply standard estimate to obtain $H^2_{loc}$. If we do not have the fact that $u \in H^2(B)$, we may hard to have the existence of $\dfrac{\partial u}{\partial n}$ on the boundary by trace theorem.

For each positive integer $n$, let $s_n$ be the alternating sum of the first $n$ primes given by $$s_n:=p_n-p_{n-1}+\ldots+(-1)^{n-1}p_1,$$ where $p_k$ denotes the $k$th prime. All the numbers $s_1,s_2,\ldots$ are pairwise distinct. The main term of $s_n$ as $n\to\infty$ is not known. It seems that $\lim_{n\to\infty}s_n/p_n=1/2$.

Here I ask a novel question involving $s_n$ on the basis of my computation.

**Question.** Is it true that
$$\{2^k3^l+s_m:\ k,l=0,1,\ldots\ \text{and}\ m=1,2,3,\ldots\}=\{2,3,\ldots\}?$$

I conjecture that the question has a positive answer. Let $r(n)$ be the number of ways to write $n$ as $2^k3^l+s_m$ with $k,l\in\{0,1,\ldots\}$ and $m\in\{1,2,\ldots\}$. The sequence $r(1),r(2),\ldots$ is available from http://oeis.org/A308411. For example, $r(2)=1$ with $2=2^03^0+p_2-p_1$. On May 25, 2019 I verified $r(n)>0$ for all $n=2,\ldots,10^6$. On May 26, 2019 Prof. Qing-Hu Hou extended the verification to $2\times 10^7$ on my request. Based on Hou's program I have verified that $r(n)>0$ for all $n=2,\ldots,10^9$.

PS: By the way, I also note that the set $$\{6^k+3^l+s_m:\ k,l=0,1,\ldots\ \text{and}\ m=1,2,\ldots\}$$ contains $3,4,\ldots,10^9$, and conjecture that this set coincides with $\{3,4,\ldots\}$ (cf. http://oeis.org/A308403).

Let $(W,H,i)$ be the classical Wiener space where $W=C_0([0,1])$, $H$ is the Cameron-Martin space. Let $A= I_{W}+a$ such that $A:W \rightarrow W$ and $a \in L^{0}(\mu,H)$, where $\mu$ is the Wiener measure. Now, clearly the most general case that we have is that $\mathbb{E}[\rho(-\delta a)]\leq1$ all the time where $\rho$ is the Wick exponential and $\delta$ is the adjoint of the weak (sobolev) derivative. I've seen that for any bounded measurable $f$, $\mathbb{E}[f\circ A]=\mathbb{E}[FL]$, where $L$ is the Radon-Nikodym derivative of $A\mu =\mu(A^{-1}(.)$ with respect to $\mu$. Then $\mathbb{E}[fL] \geq \mathbb{E}[f \circ A L \circ A \rho(-\delta a)]$, where the euqality only hold if $\mathbb{E}[\rho(-\delta a)]=1$ almost surely. How does this inequality exactly follows from Girsanov's theorem ? In the case when $\mathbb{E}[\rho(-\delta a)]=1$ it is directly comes from the definition of Girsanov's theorem but I could not deduce the inequality when this is not the case. It appears in page 9 of this paper : https://arxiv.org/pdf/0903.3891.pdf