Let G be an abelian group, let S be a sequence of terms s1, s2, . . . , sn ∈ G not all contained in a coset of a proper subgroup of G, and let W be a sequence of n consecutive integers. Let W ̄ S = {w1s1 + . . . + wnsn : wi a term of W, wi 6= wj for i 6= j}, which is a particular kind of weighted restricted sumset. We show that |W ̄S| ≥ min{|G| − 1, n}, that W ̄ S = G if n ≥ |G| + 1, and also c...

Some patterns cannot be avoided ad infinitum. A well-known example of such a pattern is an arithmetic progression in partitions of natural numbers. We observed that in order to avoid arithmetic progressions, other patterns emerge. A visualization is presented that reveals these patterns. We capitalize on the observed patterns by constructing techniques to avoid arithmetic progressions. More for...

‎we obtain the asymptotic expansion of the sequence with general term $frac{a_n}{g_n}$‎, ‎where $a_n$ and $g_n$ are the arithmetic and geometric means of the numbers $d(1),d(2),dots,d(n)$‎, ‎with $d(n)$ denoting the number of positive divisors of $n$‎. ‎also‎, ‎we obtain some explicit bounds concerning $g_n$ and $frac{a_n}{g_n}$.

Let U (n) denote the maximal length arithmetic progression in a non-uniform random subset of {0, 1} n , where 1 appears with probability pn. By using dependency graph and Stein-Chen method, we show that U (n) − cn ln n converges in law to an extreme type distribution with ln pn = −2/cn. Similar result holds for W (n) , the maximal length aperiodic arithmetic progression (mod n). An arithmetic p...

In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers n and k, the expression aw([n], k) denotes the smallest number of colors with which the integers {1, . . . , n} can be colored and still guarantee there is a rainbow arithmetic progression of length k. We establish that aw([n], 3) = Θ(log n) and aw([n...

We study an extension of the triangle removal lemma of Ruzsa and Szemerédi [Triple systems with no six points carrying three triangles, Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, NorthHolland, Amsterdam, 1978, pp. 939–945], which gave rise to a purely combinatorial proof of the fact that sets of integers of positive upper density contain three-term arithmetic progr...

Let p(n) denote the number of partitions of a non-negative integer n. A well-known conjecture asserts that every arithmetic progression contains infinitely many integers M for which p(M) is odd, as well as infinitely many integers N for which p(N) is even (see Subbarao [23]). In this paper we prove that there indeed are infinitely many integers N in every arithmetic progression for which p(N) i...

Let $\epsilon > 0$ be sufficiently small and let $0 < \eta 1/522$. We show that if $X$ is large enough in terms of $\epsilon$ then for any squarefree integer $q \leq X^{196/261-\epsilon}$ $X^{\eta}$-smooth one can obtain an asymptotic formula with power-saving error term the number integers arithmetic progression $a \pmod{q}$, $(a,q) = 1$. In case squarefree, smooth moduli this improves upon pr...