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A formula for the number of solutions of a restricted linear congruence

K. Vishnu Namboothiri (2021)

Mathematica Bohemica

Consider the linear congruence equation x 1 + ... + x k ≡ b ( mod n s ) for b ∈ ℤ , n , s ∈ ℕ . Let ( a , b ) s denote the generalized gcd of a and b which is the largest l s with l ∈ ℕ dividing a and b simultaneously. Let d 1 , ... , d τ ( n ) be all positive divisors of n . For each d j ∣ n , define 𝒞 j , s ( n ) = { 1 ≤ x ≤ n s : ( x , n s ) s = d j s } . K. Bibak et al. (2016) gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on x i . We generalize their result with generalized gcd restrictions on x i and prove that for the above linear congruence, the number of solutions...

Arithmetic progressions in sumsets

Imre Z. Ruzsa (1991)

Acta Arithmetica

1. Introduction. Let A,B ⊂ [1,N] be sets of integers, |A|=|B|=cN. Bourgain [2] proved that A+B always contains an arithmetic progression of length e x p ( l o g N ) 1 / 3 - ε . Our aim is to show that this is not very far from the best possible. Theorem 1. Let ε be a positive number. For every prime p > p₀(ε) there is a symmetric set A of residues mod p such that |A| > (1/2-ε)p and A + A contains no arithmetic progression of length (1.1) e x p ( l o g p ) 2 / 3 + ε . A set of residues can be used to get a set of integers in an obvious way. Observe...

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