More Like this

1. On a Bohr set analogue of Chowla’s conjecture

   https://doi.org/10.1007/s00209-025-03770-2  

   Teräväinen, Joni; Walker, Aled (May 2025, Mathematische Zeitschrift)

   Abstract Let$$\lambda $$ $\lambda$denote the Liouville function. We show that the logarithmic mean of$$\lambda (\lfloor \alpha _1n\rfloor )\lambda (\lfloor \alpha _2n\rfloor )$$ $\lambda \left(\lfloor {\alpha }_{1}n\rfloor \right)\lambda \left(\lfloor {\alpha }_{2}n\rfloor \right)$is 0 whenever$$\alpha _1,\alpha _2$$ ${\alpha }_1,{\alpha }_2$are positive reals with$$\alpha _1/\alpha _2$$ ${\alpha }_1/{\alpha }_2$irrational. We also show that for$$k\geqslant 3$$ $k⩾3$the logarithmic mean of$$\lambda (\lfloor \alpha _1n\rfloor )\cdots \lambda (\lfloor \alpha _kn\rfloor )$$ $\lambda \left(\lfloor {\alpha }_{1}n\rfloor \right)\cdots \lambda \left(\lfloor {\alpha }_{k}n\rfloor \right)$has some nontrivial amount of cancellation, under certain rational independence assumptions on the real numbers$$\alpha _i.$$ ${\alpha }_i.$Our results for the Liouville function generalise to produce independence statements for general bounded real-valued multiplicative functions evaluated at Beatty sequences. These results answer the two-point case of a conjecture of Frantzikinakis (and provide some progress on the higher order cases), generalising a recent result of Crnčević–Hernández–Rizk–Sereesuchart–Tao. As an ingredient in our proofs, we establish bounds for the logarithmic correlations of the Liouville function along Bohr sets. 

   more » « less
2. Field Theory of Interacting Boundary Gravitons

   https://doi.org/10.21468/SciPostPhys.13.2.038  

   Ebert, Stephen; Hijano, Eliot; Kraus, Per; Monten, Ruben; Myers, Richard M. (January 2022, SciPost Physics)

   Pure three-dimensional gravity is a renormalizable theory with twofree parameters labelled byG $G$and\Lambda $\Lambda$.As a consequence, correlation functions of the boundary stress tensor inAdS_3 ${}_3$are uniquely fixed in terms of one dimensionless parameter, which is thecentral charge of the Virasoro algebra. The same argument implies thatAdS_3 ${}_3$gravity at a finite radial cutoff is a renormalizable theory, but nowwith one additional parameter corresponding to the cutoff location. Thistheory is conjecturally dual to aT\overline{T} $T\overline{T}$-deformedCFT, assuming that such theories actually exist. To elucidate this, westudy the quantum theory of boundary gravitons living on a cutoff planarboundary and the associated correlation functions of the boundary stresstensor. We compute stress tensor correlation functions to two-loop order(G $G$being the loop counting parameter), extending existing tree levelresults. This is made feasible by the fact that the boundary gravitonaction simplifies greatly upon making a judicious field redefinition,turning into the Nambu-Goto action. After imposing Lorentz invariance,the correlators at this order are found to be unambiguous up to a singleundetermined renormalization parameter. 

   more » « less
3. Competition-driven eco-evolutionary feedback reshapes bacteriophage lambda’s fitness landscape and enables speciation

   https://doi.org/10.1038/s41467-024-45008-5  

   Doud, Michael B.; Gupta, Animesh; Li, Victor; Medina, Sarah J.; De La Fuente, Caesar A.; Meyer, Justin R. (January 2024, Nature Communications)

   Abstract A major challenge in evolutionary biology is explaining how populations navigate rugged fitness landscapes without getting trapped on local optima. One idea illustrated by adaptive dynamics theory is that as populations adapt, their newly enhanced capacities to exploit resources alter fitness payoffs and restructure the landscape in ways that promote speciation by opening new adaptive pathways. While there have been indirect tests of this theory, to our knowledge none have measured how fitness landscapes deform during adaptation, or test whether these shifts promote diversification. Here, we achieve this by studying bacteriophage$$\lambda$$ $\lambda$, a virus that readily speciates into co-existing receptor specialists under controlled laboratory conditions. We use a high-throughput gene editing-phenotyping technology to measure$$\lambda$$ $\lambda$’s fitness landscape in the presence of different evolved-$$\lambda$$ $\lambda$competitors and find that the fitness effects of individual mutations, and their epistatic interactions, depend on the competitor. Using these empirical data, we simulate$$\lambda$$ $\lambda$’s evolution on an unchanging landscape and one that recapitulates how the landscape deforms during evolution.$$\lambda$$ $\lambda$heterogeneity only evolves in the shifting landscape regime. This study provides a test of adaptive dynamics, and, more broadly, shows how fitness landscapes dynamically change during adaptation, potentiating phenomena like speciation by opening new adaptive pathways. 

   more » « less
4. Curvature and sharp growth rates of log-quasimodes on compact manifolds

   https://doi.org/10.1007/s00222-025-01315-2  

   Huang, Xiaoqi; Sogge, Christopher D (March 2025, Inventiones mathematicae)

   Abstract We obtain new optimal estimates for the$$L^{2}(M)\to L^{q}(M)$$ $L^2\left(M\right)\to L^q\left(M\right)$,$$q\in (2,q_{c}]$$ $q\in (2,q_c]$,$$q_{c}=2(n+1)/(n-1)$$ $q_c=2(n+1)/(n-1)$, operator norms of spectral projection operators associated with spectral windows$$[\lambda ,\lambda +\delta (\lambda )]$$ $[\lambda ,\lambda +\delta (\lambda \left)\right]$, with$$\delta (\lambda )=O((\log \lambda )^{-1})$$ $\delta \left(\lambda \right)=O\left({(log\lambda )}^{-1}\right)$on compact Riemannian manifolds$$(M,g)$$ $(M,g)$of dimension$$n\ge 2$$ $n\ge 2$all of whose sectional curvatures are nonpositive or negative. We show that these two different types of estimates are saturated on flat manifolds or manifolds all of whose sectional curvatures are negative. This allows us to classify compact space forms in terms of the size of$$L^{q}$$ $L^q$-norms of quasimodes for each Lebesgue exponent$$q\in (2,q_{c}]$$ $q\in (2,q_c]$, even though it is impossible to distinguish between ones of negative or zero curvature sectional curvature for any$$q>q_{c}$$ $q>q_c$. 

   more » « less
5. Wave optics of imaging with contact ball lenses

   https://doi.org/10.1038/s41598-023-32826-8  

   Maslov, A V; Jin, B; Astratov, V N (December 2023, Scientific Reports)

   Abstract Recent progress in microspherical superlens nanoscopy raises a fundamental question about the transition from super-resolution properties of mesoscale microspheres, which can provide a subwavelength resolution$$\sim \lambda /7$$ $\sim \lambda /7$, to macroscale ball lenses, for which the imaging quality degrades because of aberrations. To address this question, this work develops a theory describing the imaging by contact ball lenses with diameters$$30 $30<D/\lambda <4000$covering this transition range and for a broad range of refractive indices$$1.3<2.1$$ $1.3<n<2.1$. Starting from geometrical optics we subsequently proceed to an exact numerical solution of the Maxwell equations explaining virtual and real image formation as well as magnificationMand resolution near the critical index$$n\approx 2$$ $n\approx 2$which is of interest for applications demanding the highestMsuch as cellphone microscopy. The wave effects manifest themselves in a strong dependence of the image plane position and magnification on$$D/\lambda $$ $D/\lambda$, for which a simple analytical formula is derived. It is demonstrated that a subwavelength resolution is achievable at$$D/\lambda \lesssim 1400$$ $D/\lambda \lesssim 1400$. The theory explains the results of experimental contact-ball imaging. The understanding of the physical mechanisms of image formation revealed in this study creates a basis for developing applications of contact ball lenses in cellphone-based microscopy. 

   more » « less