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The three gap theorem asserts that for any real $\alpha $ and any positive integer $N$, the fractional parts of the sequence $0, \alpha , 2\alpha , \cdots , (N-1)\alpha $ have at most three distinct gap lengths. In this note, we extend the three gap theorem to arbitrary orbits of interval exchange transformations, and in the process, we generalize the three gap theorem and a result by Boshernitzan.

The original definition of cluster algebras by Fomin and Zelevinsky has been categorified and generalised in several ways over the course of the past 20 years, giving rise to cluster theory. This study lead to Iyama and Yang's generalised cluster categories $\mathcal{T}/\mathcal{T}^{fd}$ coming from $n$-Calabi-Yau triples $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$. In this paper, we use some classic tools of homological algebra to give a deeper understanding of such categories $\mathcal{T}/\mathcal{T}^{fd}$. Let $k$ be a field, $n\geq 3$ an integer and $\mathcal{T}$ a $k$-linear triangulated category with a triangulated subcategory $\mathcal{T}^{fd}$ and a subcategory $\mathcal{M}=\text{add}(M)$ such that $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$ is an $n$-Calabi-Yau triple. In this paper, we prove some properties of the triangulated categories $\mathcal{T}$ and $\mathcal{T}/\mathcal{T}^{fd}$. Our first result gives a relation between the Hom-spaces in these categories, using limits and colimits. Our second result is a Gap Theorem in $\mathcal{T}$, showing when the truncation triangles split. Moreover, we apply our two theorems to present an alternative proof to a result by Guo, originally stated in a more specific setup of dg $k$-algebras $A$ and subcategories of the derived category of dg $A$-modules. This proves that $\mathcal{T}/\mathcal{T}^{fd}$ is Hom-finite and $(n-1)$-Calabi-Yau, its object $M$ is $(n-1)$-cluster tilting and the endomorphism algebras of $M$ over $\mathcal{T}$ and over $\mathcal{T}/\mathcal{T}^{fd}$ are isomorphic. Note that these properties make $\mathcal{T}/\mathcal{T}^{fd}$ a generalisation of the cluster category., Comment: 17 pages. Final accepted version to appear in the Pacific Journal of Mathematics

We obtain a sharp estimate to the norm of the traceless second fundamental form of complete hypersurfaces with constant scalar curvature immersed into a locally symmetric Riemannian manifold obeying standard curvature constraints (which includes, in particular, the Riemannian space forms with constant sectional curvature). When the equality holds, we prove that these hypersurfaces must be isoparametric with two distinct principal curvatures. Our approach involves a suitable Okumura type inequality which was introduced by Melendez (Bull Braz Math Soc 45:385–404, 2014) , corresponding to a weaker hypothesis when compared with to the assumption that these hypersurfaces have a priori at most two distinct principal curvatures.

The Three Gap Theorem states that there are at most three distinct lengths of gaps if one places $n$ points on a circle, at angles of $z, 2z, 3z, \ldots nz$ from the starting point. The theorem was first proven in 1958 by S\'os and many proofs have been found since then. In this note we show how the Three Gap Theorem can easily be deduced by using Rauzy-Veech induction., Comment: 5 pages

The three distance theorem (also known as the three gap theorem or Steinhaus problem) states that, for any given real number $\alpha$ and integer $N$, there are at most three values for the distances between consecutive elements of the Kronecker sequence $\alpha, 2\alpha,\ldots, N\alpha$ mod 1. In this paper we consider a natural generalisation of the three distance theorem to the higher dimensional Kronecker sequence $\vec\alpha, 2\vec\alpha,\ldots, N\vec\alpha$ modulo an integer lattice. We prove that in two dimensions there are at most five values that can arise as a distance between nearest neighbors, for all choices of $\vec\alpha$ and $N$. Furthermore, for almost every $\vec\alpha$, five distinct distances indeed appear for infinitely many $N$ and hence five is the best possible general upper bound. In higher dimensions we have similar explicit, but less precise, upper bounds. For instance in three dimensions our bound is 13, though we conjecture the truth to be 9. We furthermore study the number of possible distances from a point to its nearest neighbor in a restricted cone of directions. This may be viewed as a generalisation of the gap length in one dimension. For large cone angles we use geometric arguments to produce explicit bounds directly analogous to the three distance theorem. For small cone angles we use ergodic theory of homogeneous flows in the space of unimodular lattices to show that the number of distinct lengths is (a) unbounded for almost all $\vec\alpha$ and (b) bounded for $\vec\alpha$ that satisfy certain Diophantine conditions., Comment: 34 pages, 7 figures. New version: Improved bounds in Theorems 1 and 9, references added, added details of proofs of later propositions, updated conjecture for three dimensions

In this paper, inspired by Fernandez-Lopez and Garcia-Rio [11] , we shall give a new lower diameter bound for compact non-trivial shrinking Ricci solitons depending on the range of the potential function, as well as on the range of the scalar curvature. Moreover, by using a universal lower diameter bound for compact non-trivial shrinking Ricci solitons by Chu and Hu [7] and by Futaki, Li, and Li [13] , we shall provide a new sufficient condition for four-dimensional compact non-trivial shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality. Furthermore, we shall give a new lower diameter bound for compact self–shrinkers of the mean curvature flow depending on the norm of the mean curvature. We shall also prove a new gap theorem for compact self–shrinkers by showing a necessary and sufficient condition to have constant norm of the mean curvature.

Let f : ( Y , g ) → ( X , g 0 ) be a nonzero degree continuous map between compact Kahler manifolds of dimension n ≥ 2 , where g 0 has constant negative holomorphic sectional curvature. Adapting the Besson–Courtois–Gallot barycentre map techniques to the Kahler setting, we prove a gap theorem in terms of the degree of f and the diastatic entropies of ( Y , g ) and ( X , g 0 ) which extends the rigidity result proved by the author in [13] .

In this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author [L. Ni, An optimal gap theorem, Invent. Math. 189 2012, 3, 737–761]. We also prove a Liouville theorem for plurisubharmonic functions on such a manifold, which generalizes a previous result of L.-F. Tam and the first author [L. Ni and L.-F. Tam, Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature, J. Differential Geom. 64 2003, 3, 457–524] and complements a recent result of Liu [G. Liu, Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds, Duke Math. J. 165 2016, 15, 2899–2919].

In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in $\re^{n+1}$, giving some conditions under which a trantranslating soliton is a hyperplane. We also show a gap theorem for the translating soliton of hypersurfaces in $R^{n+k}$, namely, if the $L^n$ norm of the second fundamental form of the soliton is small enough, then it is a hyperplane., Comment: some results are reformulated and the monotonicity formula and volume growth are added

We prove that any $n$-dimensional closed mean convex $��$-hypersurface is convex if $��\le 0.$ This generalizes Guang's work on $2$-dimensional strictly mean convex $��$-hypersurfaces. As a corollary, we obtain a gap theorem for closed $��$-hypersurfaces with $��\le 0.$, 10 pages; minor changes suggested by the referee

We investigate groups whose Cayley graphs have poor\-ly connected subgraphs. We prove that a finitely generated group has bounded separation in the sense of Benjamini--Schramm--Tim\'ar if and only if it is virtually free. We then prove a gap theorem for connectivity of finitely presented groups, and prove that there is no comparable theorem for all finitely generated groups. Finally, we formulate a connectivity version of the conjecture that every group of type $F$ with no Baumslag-Solitar subgroup is hyperbolic, and prove it for groups with at most quadratic Dehn function., Comment: 14 pages. Changes to v2: Proof of the Theorem 1.2 shortened, Theorem 1.4 added completing the no-gap result outlined in v1

We show that there exists a universal positive constant $$\varepsilon _0 > 0$$ with the following property: let g be a positive Einstein metric on the four-sphere $$S^4$$ . If the Yamabe constant of the conformal class [g] satisfies $$\begin{aligned} Y(S^4, [g]) >\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}]) - \varepsilon _0\,, \end{aligned}$$ where $$g_{\mathbb S}$$ denotes the standard round metric on $$S^4$$ , then, up to rescaling, g is isometric to $$g_{\mathbb S}$$ . This is an extension of Gursky’s gap theorem for positive Einstein metrics on $$S^4$$ .

In this paper we prove some gap theorem for complete immersed minimal submanifold of dimension no less than six or four, depending on the codimension, in a hyperbolic space $$\mathbb {H}^{n+m}(-1)$$ . That is, we show that a high dimensional complete immersed minimal submanifold M in $$ \mathbb {H}^{n+m}(-1)$$ , is totally geodesic if the $$L^d$$ norm of |A|, for some d, on geodesic balls centered at some point $$p \in M $$ has less than quadratic growth and if either $$\sup _{x \in M} |A|^2$$ is not too large or the $$L^n$$ norm of |A| on M is finite, were, A is the second fundamental form of M.

We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem., 10 pages, grant information added, minor edits

We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n) ( scal(g(t)) - scal(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, where all the constants involved depend only on the dimension $n$. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on the same space. The above curvature estimates are proved using a gap theorem for Ricci-flatness on homogeneous spaces. The proof of this gap theorem is by contradiction and uses a local $W^{2,p}$ convergence result, which holds without symmetry assumptions., Fixed up the formulation of the weak convergence theorem in Section 2

Motivated by a previous work by Zheng and the second-named author, we study pinching constants of compact Kahler manifolds with positive holomorphic sectional curvature. In particular, we prove a gap theorem on Kahler manifolds with almost-one-half pinched holomophic sectional curvature. The proof is motivated by the work of Petersen and Tao on Riemannian manifolds with almost-quarter-pinched sectional curvature.

Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for Ricci-flat manifolds, removing the volume growth assumptions in known results., The result concerning ACyl manifolds was removed from the previous version since it can be generalized and does not depend on either Ricci soliton or Ricci-flat condition. 18 pages

A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed.

In the framework of effectively enumerable topological spaces, we introduce the notion of a partial computable function. We show that the class of partial computable functions is closed under composition, and the real-valued partial computable functions defined on a computable Polish space have a principal computable numbering. With respect to the principal computable numbering of the real-valued partial computable functions, we investigate complexity of important problems such as totality and root verification. It turns out that for some problems the corresponding complexity does not depend on the choice of a computable Polish space, whereas for other ones the corresponding choice plays a crucial role.

Usually, the Weierstras gap theorem is derived as a straightforward corollary of the Riemann–Roch theorem. Our main objective in this article is to prove the Weierstras gap theorem by following an alternative approach based on “first principles”, which does not use the Riemann– Roch formula. Having mostly applications in connection with modular functions in mind, we describe our approach for the case when the given compact Riemann surface is associated with the modular curve X0(N).

In this short note, using G\"unther's volume comparison theorem and Yokota's gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton $(M^{n},g,f)$ with sectional curvature $K(g)0$ depends only on $n, A$ and $v$, if the scalar curvature $R\leq \epsilon_{n,A,v}$, then $(M,g,f)$ is isometric to the Gaussian soliton $(\mathbb{R}^{n}, g_{E}, \frac{|x|^{2}}{4})$.

We outline the proof of the gap theorem stating that any ancient solution to the Ricci flow with Perelman's reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton must be isometric to the Euclidean space for all time. This is the main result of the author's paper [Yo].

We show that a compact half-conformally flat manifold of negative type with bounded L2 energy, sufficiently small scalar curvature, and a noncollapsing assumption has all Betti numbers bounded in terms of the L2 curvature norm. We give examples of multi-ended bubbles that disrupt attempts to improve these Betti number bounds. We show that bounded self-dual solutions of dω=0 on asymptotically locally Euclidian (ALE) manifold ends display a rate-of-decay gap: they are either asymptotically Kahler, or they have a decay rate of O(r−4) or better.

We show that there exists a positive real number $\delta>0$ such that for any normal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, that is, the minimal log discrepancy of $X$ is less than $1$, then the minimal log discrepancy of $X$ is not greater than $1-\delta$. As applications, we show that the set of all non-canonical klt Calabi-Yau $3$-folds are bounded modulo flops, and the global indices of all klt Calabi-Yau $3$-folds are bounded from above., Comment: 39 pages, comments are welcome; v2: 40 pages, slightly modified, more discussion on computation of $\delta$ added; v3: final version, to appear in J. Algebraic Geom

Let M be an n ( ≥ 7 ) -dimensional complete submanifold with parallel mean curvature in the hyperbolic space H n + p , whose mean curvature satisfies H 2 − 1 ≤ 0 . Denote by A ˚ and B R ( q ) the trace free second fundamental form of M and the geodesic ball of radius R centered at q ∈ M , respectively. We prove that if lim sup R → ∞ ∫ B R ( q ) | A ˚ | 2 d M R 2 = 0 , and if ( ∫ M | A ˚ | n d M ) 2 / n + 2 n ( n − 2 ) 3 n ( n − 1 ) H ( ∫ M | A ˚ | n / 2 d M ) 2 / n ≤ C ( n ) , then M is congruent to an n-dimensional hyperbolic space or the Euclidean space R n . Here C ( n ) is an explicit positive constant depending only on n. We also obtain a similar gap theorem in the case where n = 5 , 6 .

The well-known Three Gap Theorem states that there are at most three gap sizes in the sequence of fractional parts [Formula: see text]. It is known that if one averages over [Formula: see text], the distribution becomes continuous. We present an alternative approach, which establishes this averaged result and also provides good bounds for the error terms.

We show that the Gap Theorem for separated sequences by M. Mitkovski and A. Poltoratski can be deduced directly from the classical Beurling–Malliavin formula for the radius of completeness.