A contribution to infinite disjoint covering systems

Let the collection of arithmetic sequences {din+ bi : n ∈ Z}i∈I be a disjoint covering system of the integers. We prove that if di = pq for some primes p, q and integers k, l ≥ 0, then there is a j 6= i such that di|dj . We conjecture that the divisibility result holds for all moduli. A disjoint covering system is called saturated if the sum of the reciprocals of the moduli is equal to 1. The above conjecture holds for saturated systems with di such that the product of its prime factors is at most 1254. A Beatty sequence is defined by S(α, β) := {bαn + βc}n=1, where α is positive and β is an arbitrary real constant. A conjecture of Fraenkel asserts that if {S(αi, βi) : i = 1 . . .m} is a collection of m ≥ 3 Beatty sequences which partitions the positive integers then αi/αj is an integer for some i 6= j. Special cases of the conjecture were verified by Fraenkel [2], Graham [4] and Simpson [9]. For more references on Beatty sequences see [1] and [11]. If α is integral, then S(α, β) is an arithmetic sequence. Mirsky and Newman, and later independently Davenport and Rado proved that if {{ain + bi} : i = 1 . . .m} is a partition of the positive integers, then ai = aj for some i 6= j. This settles Fraenkel’s conjecture for integral α’s. We formulate a related conjecture for partitions to infinite number of arithmetic sequences. We denote by A(d, b) the arithmetic sequence {dn + b : n ∈ Z}. Let a collection S of arithmetic sequences {A(di, bi) : i ∈ I} be called a covering 52 János Barát, Péter P. Varjú system (CS), if the union of the sequences is Z. The CS is finite or infinite according to the set I. The numbers di are called the moduli of the CS. A conjecture similar to Fraenkel’s was posed by Schinzel, that for any finite CS, there is a pair of distinct indices i, j for which di|dj . This was verified by Porubský [8] assuming some extra conditions. When the sequences of a CS are disjoint, it is called a disjoint covering system (DCS). The structure of DCS’s is a wide topic of research. We only mention here a few results about IIDCS’s (such DCS’s that the number of sequences are infinite, and the moduli are distinct). For further references see [7]. There is a natural method to construct DCS’s, the following construction appeared in [10]: Example 1. Let I = N, and d1|d2|d3 . . . be positive integers. Define the bi’s recursively to be an integer of minimal absolute value not covered by the sequences A(dj , bj), (j < i). Indeed this gives a DCS; if A(di, bi) and A(dj , bj) (j < i) do intersect then A(di, bi) ⊂ A(dj , bj) as dj |di, which contradicts the definition of bi. Also the definition of bi guarantees that an integer of absolute value n is covered by one of the first 2n+ 1 sequences. If the sum of the reciprocals of the moduli equals 1, we call the DCS saturated. Apparently every finite DCS is saturated, but this property is rather ”rare” for IIDCS’s. The IIDCS in the example above is saturated only for di = 2i. Stein [10] asked whether this is the unique example. Krukenberg [5] answered this in the negative, then Fraenkel and Simpson [3] characterised all IIDCS, whose moduli are of form 2k3l. Lewis [6] proved that if a prime greater than 3 divides one of the moduli, then the set of all prime divisors of the moduli is infinite. We formulate the following conjecture: Conjecture 2. If {A(di, bi) : i ∈ I} is a DCS, then for all i there exists an index j 6= i such that di|dj. This conjecture is valid for the above examples, and also valid for those appearing in [3]. It is also known for finite DCS’s, being a consequence of Corollary 2 of [8]. We prove the following special case of Conjecture 2. Theorem 3. If {A(di, bi) : i ∈ I} is a DCS, and di = pkql for some primes p, q and integers k, l ≥ 0, then di|dj for some j 6= i. Proof. Let (a, b) denote the greatest common divisor of the integers a, b, and [a, b] their least common multiple. We will use the fact, that the sequences A(d1, b1) and A(d2, b2) are disjoint if and only if (d1, d2) x1 − x2, where xi is an arbitrary number covered by A(di, bi) for i = 1, 2. A contribution to infinite disjoint covering systems 53 If l = 0, consider the sequence A(dj , bj) that covers bi + pk−1. Then (dj , di) bi + pk−1 − bi = pk−1. Thus (dj , di) = pk = di, so di|dj which was to be proved. Now we may assume that k, l > 0. Assume to the contrary, that the theorem is false. Defining bj = bj − bi, we get another DCS {A(dj , bj) : j ∈ I}, where bi = 0. Hence we may assume that bi = 0. Let Aj = A(dj , bj)∩A(pq, 0). Either Aj is empty or an arithmetic sequence, whose modulus is [pk−1ql−1, dj ]. Let Bj = { x pk−1ql−1 ∣∣∣∣x ∈ Aj}. The nonempty sequences among the Bj ’s form a DCS. Notice that pq divides the modulus of Bj if and only if di = pq|dj . Since the modulus of Bi is pq, it remains to prove the theorem for k = l = 1. Assume di = pq, and pq dj for i 6= j. Assume that p + q is covered by the sequence A(dt, bt) of the DCS. We prove that p dt. Assume to the contrary that p|dt. Let dt = d · pm, where p d. Then (pq, d)|q, and there exist a pair of positive integers u, v such that q = pq · u − d · v. Let a = p + q + dv = p + pqu. Assume that a is covered by A(ds, bs). If A(ds, bs) and A(dt, bt) are the same sequences, then ds = dt, and p|ds. Otherwise (ds, dt) p+q+dv−p−q = dv, which yields p|ds. Since pq a, s 6= i, and (ds, di) p + pqu − pqu = p, thus q|ds. This contradicts di ds. Similar argument shows q dt, which contradicts (di, dt) bi − bt. So the proof is complete. As a byproduct of the previous proof, we got the following lemma: Lemma 4. Suppose there is a DCS {A(di, bi) : i ∈ I} and an index i ∈ I such, that di = p1 1 p α2 2 . . . p αk k , where the p’s are distinct primes, and di dj for all j 6= i. Then there exists another DCS {A(d̂i, b̂i) : i ∈ Î} and an index î ∈ Î such that d̂î = p1p2 . . . pk and d̂î d̂j for all j 6= î. So it is sufficient to verify the conjecture for square-free moduli. When di has more than two different prime factors, the situation seems to be much more complicated. We can still say something for saturated DCS’s. We need the following concepts: Suppose A ⊆ Z. Let Sn(A) = |{x ∈ A : −n < x < n}| be the number of elements of A with absolute value less than n. We define the density of A to be d(A) = lim n→∞ Sn(A) 2n−1 if the limit exists, and in that case we say, that the density of A exists. We will use following facts. The density is finitely additive, and the density of arithmetic sequences exist, and d(A(d, b)) = 1 d . Let {A(di, bi) : i ∈ I} be a saturated DCS, and J ⊆ I. Lemma 2.2 of [6] states, that the density of X := ⋃ j∈J A(dj , bj) exists, and d(X) = ∑