Sukumar Das Adhikari

Transcendental infinite sums

Abstract We show that it follows from results on linear forms in logarithms of algebraic numbers such as where χ is any non-principal Dirichlet character and (Fn∞n=0 the Fibonacci sequence, are transcendental.

On a question regarding visibility of lattice points--III

For a positive integer m, let ω(m) denote the number of distinct prime factors of m. Let h(n) be a function defined on the set of positive integers such that h(n) → ∞ as n → ∞ and let En(h) = {d:d is a positive integer, d ≤ n, ω(d) ≥ h(n)}. Writing Δn = {(x,y):x,y are integers, 1 ≤ x, y ≤ n}, in the present paper we show that one can give explicit description of a set Xn ⊂ Δn such that Δn is visible from Xn with at most 100|En(h)|2 exceptional points and for all sufficiently large n, one has Xn| ≤ 800h(n)log log h(n).As a corollary it follows that one can give explicit description of a set Yn ⊂ Δn such that for large ns, Δn is visible except for at most 100n2(log log n)2 exceptional points from Yn where Yn satisfies |Yn| = O((log logn)(log log log log n)).

Remarks on some zero-sum problems

Abstract Let Z denote the ring of integers and for a prime p and positive integers r and d, let fr(P, d) denote the smallest positive integer such that given any sequence of fr(p, d) elements in (Z/pZ(d, there exists a subsequence of (rp) elements whose sum is zero in (Z/pZ(d. That f1(p, 1) = 2p − 1, is a classical result due to Erdős, Ginzburg and Ziv. Whereas the determination of the exact value of f1(p, 2) has resisted the attacks of many well known mathematicians, we shall see that exact values of fr(p, 1) for r ≥ 1 can be easily obtained from the above mentioned theorem of Erdős, Ginzburg and Ziv and those of fr(p, 2) for r ≥ 2 can be established by the existing techniques developed by Alon, Dubiner and Ronyai in connection with obtaining good upper bounds for f1(p, 2). We shall also take this opportunity to describe some of the early results in the introduction.

On a theorem of Graham

Abstract A strengthened form of Gurevichs conjecture was proved by R. L. Graham, which says that for any α > 0 and any pair of non-parallel lines L1 and L2, in any partition of the plane into finitely many classes, some class contains the vertices of a triangle which has area α and two sides parallel to the lines Li. In this note, using the main idea of Graham, we present a shorter proof of the result.

On an error term related to the Jordan totient function Jk(n)

Abstract We investigate the error terms E k (x)= ∑ n⩽x J k (n)− x k+1 (k+1)ζ(k+1) for k⩾2 , where J k (n) = n k Π p|n (1 − 1 p k ) for k ≥ 1. For k ≥ 2, we prove ∑ n⩽x E k (n)∼ x k+1 2(k+1)ζ(k+1) . Also, lim inf n→∞ E k (x) x k ⩽ D ζ(k+1) , where D = .7159 when k = 2, .6063 when k ≥ 3. On the other hand, even though lim inf n→∞ E k (x) x k ⩽− 1 2ζ(k+1) , Ek(n) > 0 for integers n sufficiently large.

The polynomial method in the study of zero-sum theorems

Here we try to have a glimpse of the polynomial method used in the study of some zero-sum problems in additive combinatorics. While the first few sections of this paper are of an expository nature, in the last section we present some new results on bounds of some zero-sum constants obtained by generalizing a polynomial method. We also mention some open questions.

Number of Weighted Subsequence Sums with Weights in {1, –1}

Abstract Let G be an abelian group of order n of the form G ≅ ℤ n 1 ⊕ ℤ n 2 ⊕ ⋯ ⊕ ℤ nr , where ni | ni +1 for 1 ≤ i < r and n 1 > 1. Let A = {1, –1}. Given a sequence S with elements in the given group G and of length n + k such that the natural number k satisfies , where r′ = |{i ∈ {1, 2, …, r} : 2 | ni }|, if S does not have an A-weighted zero-sum subsequence of length n, we obtain a lower bound on the number of A-weighted n-sums of the sequence S. This is a weighted version of a result of Bollobás and Leader. As a corollary, one obtains a result of Adhikari, Chen, Friedlander, Konyagin and Pappalardi. A result of Yuan and Zeng on the existence of zero-smooth subsequences and the DeVos–Goddyn–Mohar Theorem are some of the main ingredients of our proof.

On the degree of regularity of a certain quadratic Diophantine equation

Abstract We show that, for every positive integer r , there exists an integer b = b ( r ) such that the 4-variable quadratic Diophantine equation ( x 1 − y 1 ) ( x 2 − y 2 ) = b is r -regular. Our proof uses Szemeredi’s theorem on arithmetic progressions.

Equation-regular sets and the Fox–Kleitman conjecture

Abstract Given k ≥ 1 , the Fox–Kleitman conjecture from 2006 states that there exists a nonzero integer b such that the 2 k -variable linear Diophantine equation ∑ i = 1 k ( x i − y i ) = b is ( 2 k − 1 ) -regular. This is best possible, since Fox and Kleitman showed that for all b ≥ 1 , this equation is not 2 k -regular. While the conjecture has recently been settled for all k ≥ 2 , here we focus on the case k = 3 and determine the degree of regularity of the corresponding equation for all b ≥ 1 . In particular, this independently confirms the conjecture for k = 3 . We also briefly discuss the case k = 4 .

On the finiteness of some n-color Rado numbers

Abstract For integers k , n , c with k , n ≥ 1 , the n -color Rado number R k ( n , c ) is defined to be the least integer N if any, or infinity otherwise, such that for every n -coloring of the set { 1 , 2 , … , N } , there exists a monochromatic solution in that set to the linear equation x 1 + x 2 + ⋯ + x k + c = x k + 1 . A recent conjecture of ours states that R k ( n , c ) should be finite if and only if every divisor d ≤ n of k − 1 also divides c . In this paper, we complete the verification of this conjecture for all k ≤ 7 . As a key tool, we first prove a general result concerning the degree of regularity over subsets of Z of some linear Diophantine equations.