Results on measure-universal sets, dyadic grids, random routing and periodic hitting sets, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Finite sets are measure universal — Proposition 1.2. Every measurable set A of positive measure contains x + sF for every finite set F and every 0 < s <= s_0, for some s_0 > 0. No hypothesis.
• Geometric progressions are not measure universal (OpenAI) — Theorem 2.2. For every q in (0,1) and eta in (0,1) there is a compact E in [0,1] with m(E) > 1 - eta containing no affine copy x + s{q^n : n >= 1}, s != 0. No hypothesis.
• Periodic hitting sets — Proposition 3.1. For every p in (0,1) there is an open 1-periodic set H of density at most 6p such that every x + t q^n, x real, t in [1,2], meets H for some n >= 1; proved in the fourth lesson. No hypothesis.
• Nested dyadic grids — Lemma 1.1. N_b = 2^ceil(log2(c q^-b)) satisfies c q^-b <= N_b < 2c q^-b with c = 4/(1-q); finer keys determine coarser keys; keys are constant on segments without grid points. No hypothesis.
• Spans of windows on an ordered tree — Lemma 2.1. The windows below a vertex of height h occupy sigma_h consecutive integers, and the block of an edge leaving height h spans at most 2 r_h integers. No hypothesis.
• Stable centers keep earlier keys — Lemma 3.2. The unstable centers have density at most 4Kc q^(g+1), and at a stable center the points x + t q^n of a window keep the keys of x in all earlier grids. No hypothesis.
• Separation within a window — Lemma 4.1. The points x and x + t q^n, n in one window, have pairwise distinct keys at the window's resolution and every finer one. No hypothesis.
• Mean density of the routed random set — Lemma 1.1. The random 1-periodic set B defined by routing points through selector tables to terminal tables of Bernoulli(p) bits has expected density p. No hypothesis.
• Routes without a default choice — Lemma 2.1. The route of a fixed center makes no default choice with probability (1 - 2^(1-M))^d. No hypothesis.
• The route is preserved at stable centers — Lemma 3.1. At a stable center, the points of the windows below the first default vertex U follow the route of the center to U, and a successful local test puts them in B. No hypothesis.
• Independent tests at a fixed scale — Lemma 4.1. Conditionally on the exposure of a stable center with a default choice, all (M-1) r local tests at a fixed dilation fail with probability exactly (1 - p/2)^((M-1) r). No hypothesis.
• All dilations at a stable center — Lemma 1.1. Conditionally on such an exposure, some dilation t in [1,2] misses B at all window indices with probability at most D_{M,q}(r) e^(-p(M-1)r/2). No hypothesis.
• Misses at stable centers — Lemma 3.1. With the chosen parameters, a stable center is missed at some dilation with probability at most 2p. No hypothesis.

Reading order

Four lessons. The first states the problem and the theorem and reduces it to a periodic hitting set; the second and third set up the grids, the tree of windows and the random routing; the fourth proves the hitting set and with it the theorem. Everything used is proved in the course apart from basic facts of Lebesgue measure and real analysis from the core courses.