Distribution of semi-$k$-free integers

Ergodic Properties of k-Free Integers in Number Fields

Let K/Q be a degree d extension. Inside the ring of integers OK we define the set of k-free integers Fk and a natural OK-action on the space of binary OK-indexed sequences, equipped with an OK-invariant probability measure associated to Fk. We prove that this action is ergodic, has pure point spectrum, and is isomorphic to a Zdaction on a compact abelian group. In particular, it is not weakly m...

ON THE SCHNIRELMANN DENSITY OF k-FREE INTEGERS

Sum-free Sets of Integers

A set S of integers is said to be sum-free if a, b e 5 implies a + b 6 S. In this paper, we investigate two new problems on sum-free sets: (1) Let f(k) denote the largest positive integer for which there exists a partition of (1, 2,... ,f(k)) into k sum-free sets, and let h(k) denote the largest positive integer for which there exists a partition of {1, 2, . . . , h(k)) into k sets which are su...

Logarithmic Representability of Integers as K-sums

A set A = Ak,n ⇢ [n] [ {0} is said to be an additive k-basis if each element in {0, 1, . . . , kn} can be written as a k-sum of elements of A in at least one way. Seeking multiple representations as k-sums, and given any function (n) ! 1, we say that A is said to be a truncated (n)-representative k-basis for [n] if for each j 2 [↵n, (k ↵)n] the number of ways that j can be represented as a k-su...

Hermitian K-theory of the Integers

Rognes andWeibel used Voevodsky’s work on the Milnor conjecture to deduce the strong Dwyer-Friedlander form of the Lichtenbaum-Quillen conjecture at the prime 2. In consequence (the 2-completion of) the classifying space for algebraic K-theory of the integers Z[1/2] can be expressed as a fiber product of wellunderstood spaces BO and BGL(F3) over BU . Similar results are now obtained for Hermiti...