We improve the quantitative estimate for Roth’s theorem on threeterm arithmetic progressions, showing that if A ⊂ {1, . . . , N} contains no nontrivial three-term arithmetic progressions then |A| N(log logN)4/ logN . By the same method we also improve the bounds in the analogous problem over Fq [t] and for the problem of finding long arithmetic progressions in a sumset.

In this paper, we shall establish a hierarchy of functional equations (as G-function hierarchy) by unifying zeta-functions that satisfy the Hecke equation and those corresponding to Maass forms in framework ramified with (essentially) two gamma factors through Fourier–Whittaker expansion. This unifies theory Epstein associated sense gives method construction forms. long term, is remote conseque...

Let p(n) denote the number of partitions of a non-negativeinteger n. A well-known conjecture asserts that every arithmetic progression contains innnitely many integers M for which p(M) is odd, as well as innnitely many integers N for which p(N) is even (see Subbarao 22]). From the works of various authors, this conjecture has been veriied for every arithmetic progression with modulus t when t H...

A 3–coloring of an abelian group G is rainbow–free if there is no 3–term arithmetic progression with its members having pairwise distinct colors. We describe the structure of rainbow–free colorings of abelian groups. This structural description proves a conjecture of Jungić et al. on the size of the smallest chromatic class of a rainbow–free coloring of cyclic groups.

Let W (3, k) denote the largest integer w such that there is a red/blue coloring of {1, 2, . . . , w} which has no red 3-term arithmetic progression and no block of k consecutive blue integers. We show that for some absolute constant c, W (3, k) ≥ k log k for all k. AMS Mathematics Subject Classification: 05D10

Consider a coloring of {1, 2, . . . , n} in 3 colors, where n ≡ 0 (mod 3). If all the color classes have the same cardinality, then there is a 3-term arithmetic progression whose elements are colored in distinct colors. This rainbow variant of van der Waerden’s theorem proves the conjecture of the second author.

We show that the abc conjecture implies that the number of terms of any arithmetic progression consisting of almost perfect ”inhomogeneous” powers is bounded, moreover, if the exponents of the powers are all ≥ 4, then the number of such progressions is finite. We derive a similar statement unconditionally, provided that the exponents of the terms in the progression are bounded from above.

Assuming a conjectural upper bound for the least prime in an arithmetic progression, we show that n-bit integers may be multiplied in O(n logn 4 ∗ n) bit operations.