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We prove the following conjecture of Furstenberg (1969): if 𝐴,𝐵⊂[0,1] are closed and invariant under ×𝑝 mod1 and ×𝑞 mod1, respectively, and if log𝑝/log𝑞∉ℚ, then for all real numbers 𝑢 and 𝑣, dim_H (𝑢𝐴 + 𝑣) ∩ 𝐵 ≤ max {0, dim_H 𝐴 + dim_H 𝐵 − 1}. We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on ℝ. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of planar self-similar sets satisfying SSC and certain natural irreducible conditions.

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This paper gives a new and direct construction of the multi-prime big de Rham–Witt complex, which is defined for every commutative and unital ring; the original construction by Madsen and myself relied on the adjoint functor theorem and accordingly was very indirect. The construction given here also corrects the 2-torsion which was not quite correct in the original version. The new construction is based on the theory of modules and derivations over a$λ$-ring which is developed first. The main result in this first part of the paper is that the universal derivation of a$λ$-ring is given by the universal derivation of the underlying ring together with an additional structure depending directly on the$λ$-ring structure in question. In the case of the ring of big Witt vectors, this additional structure gives rise to divided Frobenius operators on the module of Kähler differentials. It is the existence of these divided Frobenius operators that makes the new construction of the big de Rham–Witt complex possible. It is further shown that the big de Rham–Witt complex behaves well with respect to étale maps, and finally, the big de Rham–Witt complex of the ring of integers is explicitly evaluated.

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