#Lattice math set theory mod
Suppose p = 2 or p is a prime congruent to 3 mod 4, and p divides b or d. For, suppose r = a/b, s = c/d (written in lowest terms), and r 2 + s 2 = e ∈ ℤ. Note that if (r, s) ∈ ℚ × ℚ and r 2 + s 2 ∈ ℤ, then (r, s) ∈ H. Let H be the subgroup of ℚ × ℚ consisting of elements with denominators only divisible by primes of the form 4k + 1. When constructing the selector we only have to worry about points that are joined. The set S is the image set of the selector.įor this we define a graph on the Abelian group ℚ/ℤ × ℚ/ℤ by joining x with y if and only if there are elements g(x) ∈ x + (ℤ × ℤ) and g(y) ∈ y + (ℤ × ℤ) such that the square of the distance between g(x) and g(y) is an integer. We want to show there is a map f : (ℚ/ℤ) × (ℚ/ℤ) → ℚ × ℚ such that ρ 2(f(x), f(y)) ∉ ℤ holds for x ≠ y and f is a selector: f(, ) ∈ ×. Our methods allow us to prove a strengthening of Theorem 1. Let us say a lattice distance is a real number of the form, where n, m ∈ ℤ. We avoid this by using a hull construction that we describe shortly. 4.Ī straightforward induction argument quickly runs into problems, as noted in refs. Steinhaus's problem and variants were discussed in some detail by Croft ( 7) and have been updated in sections E10 and G9 of ref. Kolountzakis and Wolff ( 9) showed that there is no measurable Steinhaus set for the higher-dimensional version of Steinhaus's problem for the standard lattice.
This problem has been the origin of many papers including those of Beck ( 6), Croft ( 7), Komjáth ( 3), and Kolountzakis ( 8). We call a set S as in Theorem 1 a Steinhaus set and note that whether there can be a Lebesgue measurable Steinhaus set remains unsolved. There is a set S ⊆ ℝ 2 such that for every isometric copy L of the integer lattice ℤ 2 we have |S ∩ L| = 1. Here by using a combination of techniques from analysis, set theory, number theory, and plane geometry we show the answer is in the affirmative.Theorem 1. 4 and 5) but has remained unsolved until now. This specific problem has been widely noted (see, e.g., refs. Steinhaus also asked several related questions that have been stated and studied in refs. Is there a set S in the plane such that every set congruent to ℤ 2 has exactly one point in common with S? The problem seems to have first appeared in a 1958 paper of Sierpiński ( 1).
, 11 (1980) pp.In the 1950s, Steinhaus posed the following problem. Pudlák, "Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups" Alg. Kronheimer (ed.), Aspects of Topology: in Memory of Hugh Dowker, Lect. Isbell, "Graduation and dimension in locales" I.H. Galián, "Theoriá de la dimensión", Madrid (1979) Feit, "An interval in the subgroup lattice of a finite group which is isomorphic to $M_7$" Alg. Ore the latter used the term "structure" instead of "lattice", but this quickly became obsolete except in Russia, where it survived until the 1960-s. The development of the subject in the 1930-s was largely the work of G. The first significant work on lattices was done by E. 1) A linearly ordered set (or chain) $ M $Ģ) The subspaces of a vector space ordered by inclusion, where
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