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Creators/Authors contains: "Tao, Terence"

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  1. ; (, Journal of the European Mathematical Society)
    In the 1960s, Berger famously showed that translational tilings of\mathbb{Z}^{2}with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability oftranslational monotilings(tilings by translations of a single tile) in\mathbb{Z}^{2}. The decidability of translational monotilings in higher dimensions remained unsolved. In this paper, by combining our recently developed techniques with ideas introduced by Aanderaa–Lewis, we finally settle this problem, achieving the undecidability of translational monotilings of (periodic subsets of) virtually\mathbb{Z}^{2}spaces, namely, spaces of the form\mathbb{Z}^{2}\times G_{0}, whereG_{0}is a finite Abelian group. This also implies the undecidability of translational monotilings in\mathbb{Z}^{d},d\geq 3. 
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    Free, publicly-accessible full text available June 21, 2026
  2. (, Notices of the American Mathematical Society)
    Free, publicly-accessible full text available January 1, 2026
  3. ; ; (, Random Structures & Algorithms)
    Abstract The entropic doubling of a random variable taking values in an abelian group is a variant of the notion of the doubling constant of a finite subset of , but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of Pálvölgyi and Zhelezov on the “skew dimension” of subsets of with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of with small doubling; (3) A proof that the Polynomial Freiman–Ruzsa conjecture over implies the (weak) Polynomial Freiman–Ruzsa conjecture over . 
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  4. ; (, Annals of Mathematics)
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  5. ; ; (, Proceedings of the London Mathematical Society)
    Abstract We give an improved lower bound for the average of the Erdős–Hooley function , namely for all and any fixed , where is an exponent previously appearing in work of Green and the first two authors. This improves on a previous lower bound of of Hall and Tenenbaum, and can be compared to the recent upper bound of of the second and third authors. 
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  6. (, Communications of the American Mathematical Society)
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  7. ; ; (, Communications of the American Mathematical Society)
    Let Γ<#comment/> \Gamma be a countable abelian group. An (abstract) Γ<#comment/> \Gamma -system X \mathrm {X} - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of Γ<#comment/> \Gamma - is said to be aConze–Lesigne systemif it is equal to its second Host–Kra–Ziegler factor Z 2 ( X ) \mathrm {Z}^2(\mathrm {X}) . The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian Γ<#comment/> \Gamma , namely that they are the inverse limit of translational systems G n / Λ<#comment/> n G_n/\Lambda _n arising from locally compact nilpotent groups G n G_n of nilpotency class 2 2 , quotiented by a lattice Λ<#comment/> n \Lambda _n . Results of this type were previously known when Γ<#comment/> \Gamma was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers U 3 ( G ) U^3(G) norm for arbitrary finite abelian groups G G . 
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  8. ; (, Journal of the European Mathematical Society)
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  9. ; (, Journal d'Analyse Mathématique)
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  10. ; ; (, Inventiones mathematicae)
    Abstract We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval $[1,x]$ [ 1 , x ] . Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set. 
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