Symmetrically approximately continuous functions, consistent density theorems, and Fubini type inequalities

Using the continuum hypothesis, Sierpinski constructed a non-measurable function f such that {h : f( x + h) not equal f(x - h)} is countable for every x: Clearly, such a function is symmetrically approximately continuous everywhere.Here we to show that Sierpinski's example cannot be constructed in ZFC. Moreover, we show it is consistent with ZFC that if a function is symmetrically approximately continuous almost everywhere, then it is measurable

Irregular tilings of regular polygons with similar triangles

We say that a triangle $T$ tiles a polygon $A$, if $A$ can be dissected into finitely many nonoverlapping triangles similar to $T$. We show that if $N>42$, then there are at most three nonsimilar triangles $T$ such that the angles of $T$ are rational multiples of $\pi$ and $T$ tiles the regular $N$-gon. A tiling into similar triangles is called regular, if the pieces have two angles, \al and \be, such that at each vertex of the tiling the number of angles \al is the same as that of \be. Otherwise the tiling is irregular. It is known that for every regular polygon $A$ there are infinitely many triangles that tile $A$ regularly. We show that if $N>10$, then a triangle $T$ tiles the regular $N$-gon irregularly only if the angles of $T$ are rational multiples of $\pi$. Therefore, the numbers of triangles tiling the regular $N$-gon irregularly is at most three for every $N>42$

Orders of Absolute Measurability

AbstractA subset A of the torus [0,1)k is called absolute measurable if the value of μ(A) is the same for every finitely-additive translation-invariant probability measure μ defined on all subsets of [0,1)k. We define four set functions (called orders) that measure how “strongly” a set A is absolute measurable. The order o(A) equals k−dimB(∂ A) and is connected to the Jordan measurability of A. The order δ(A) measures how small the oscillation of the average of n translates of χA can be. The order τ(A) is related to the absolute inner and outer measures defined by Tarski; finally, σ(A) is defined by the oscillation of those functions that are “scissor-congruent” to χA. We prove that o⪡δ⪡τ⪡σ, that is, each of the orders o, δ, τ, σ is “finer” than the previous one. We investigate the connection between the orders and questions of equidecomposability. We show that, under certain conditions, a set of large order is equidecomposable to a cube and present some results in the other direction as well

Magic Mirrors, Dense Diameters, Baire Category

Abstract. An old result of Zamfirescu says that for most convex curves C in the plane most points in R 2 lie on infinitely many normals to C, where most is meant in Baire category sense. We strengthen this result by showing that ‘infinitely many’ can be replaced by ‘continuum many’ in the statement. We present further theorems in the same spirit

Elementary And Integral-elementary Functions

By an integral-elementary function we mean any real function that can be obtained from the constants, sin x, e(x), log x, and arcsin x (defined on (-1, 1)) using the basic algebraic operations, composition and integration. The rank of an integral-elementary function f is the depth of the formula defining f. The integral-elementary Functions of rank less than or equal to n are real-analytic and satisfy a common algebraic differential equation P-n(f, f',..., f((k))) = 0 with integer coefficients. We prove that every continuous function f: R --> R can be approximated uniformly by integral-elementary functions of bounded rank. Consequently, there exists an algebraic differential equation with integer coefficients such that its everywhere analytic solutions approximate every continuous function uniformly. This solves a problem posed by L. A. Rubel. Using the same basic functions as above, but allowing only the basic algebraic operations and compositions, we obtain the class of elementary functions. We show that every differentiable function with a derivative not exceeding an iterated exponential can be uniformly approximated by elementary functions of bounded rank. If we include the function arcsin x defined on [-1, 1], then the resulting class of naive-elementary functions will approximate every continuous function uniformly. We also show that every sequence can be uniformly approximated by elementary functions, and that every integer sequence can be represented in the form f(n), where f is naive-elementary

A combinatorial property of cardinals

(GCH) For every cardinal kappagreater than or equal toomega(2) there exists F : [kappa](less than or equal to2)--> {0,1} such that for every f : kappa--> [kappa](<ω), i<2, there are x; y such that F(x, t) = i (t is an element of f(y)); F(u,y) =i (u is an element of f(x))

Tiling by rectangles and alternating current

This paper is on tilings of polygons by rectangles. A celebrated physical interpretation of such tilings due to R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte uses direct-current circuits. The new approach of the paper is an application of alternating-current circuits. The following results are obtained: - a necessary condition for a rectangle to be tilable by rectangles of given shapes; - a criterion for a rectangle to be tilable by rectangles similar to it but not all homothetic to it; - a criterion for a generic polygon to be tilable by squares. These results generalize the ones of C. Freiling, R. Kenyon, M. Laczkovich, D. Rinne and G. Szekeres.Comment: In English and in Russian; 21 pages; 6 figures; minor improvement of exposition, Russian translation adde

Limit theorems for self-similar tilings

We study deviation of ergodic averages for dynamical systems given by self-similar tilings on the plane and in higher dimensions. The main object of our paper is a special family of finitely-additive measures for our systems. An asymptotic formula is given for ergodic integrals in terms of these finitely-additive measures, and, as a corollary, limit theorems are obtained for dynamical systems given by self-similar tilings.Comment: 36 pages; some corrections and improved exposition, especially in Section 4; references adde