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Abstract We study the volume growth of metric balls as a function of the radius in discrete spaces and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called Ollivier curvature and discuss similar results under other types of discrete Ricci curvature. Following recent work in the continuous setting of Riemannian manifolds (by the 1st author), we then bound the eigenvalues of the Laplacian of a graph under bounds on the volume growth. In particular, $$\lambda _2$$ of the graph can be bounded using a weighted discrete Hardy inequality and the higher eigenvalues of the graph can be bounded by the eigenvalues of a tridiagonal matrix times a multiplicative factor, both of which only depend on the volume growth of the graph. As a direct application, we relate the eigenvalues to the Cheeger isoperimetric constant. Using these methods, we describe classes of graphs for which the Cheeger inequality is tight on the 2nd eigenvalue (i.e. the 1st nonzero eigenvalue). We also describe a method for proving Buser’s Inequality in graphs, particularly under a lower bound assumption on curvature.
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This content will become publicly available on February 17, 2026
Joint ergodicity for functions of polynomial growth
Abstract We provide necessary and sufficient conditions for joint ergodicity results for systems of commuting measure preserving transformations for an iterated Hardy field function of polynomial growth. Our method builds on and improves recent techniques due to Frantzikinakis and Tsinas, who dealt with multiple ergodic averages along Hardy field functions; it also enhances an approach introduced by the authors and Ferré Moragues to study polynomial iterates. The more general expression, in which the iterate is a linear combination of a Hardy field function of polynomial growth and a tempered function, is studied as well.
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- Award ID(s):
- 2247331
- PAR ID:
- 10587129
- Publisher / Repository:
- Springer Nature SharedIt
- Date Published:
- Journal Name:
- Israel Journal of Mathematics
- ISSN:
- 0021-2172
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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