Binary linear forms over finite sets of integers
نویسندگان
چکیده
منابع مشابه
Binary Linear Forms over Finite Sets of Integers
Let A be a finite set of integers. For a polynomial f(x1, . . . , xn) with integer coefficients, let f(A) = {f(a1, . . . , an) : a1, . . . , an ∈ A}. In this paper it is proved that for every pair of normalized binary linear forms f(x, y) = u1x + v1y and g(x, y) = u2x + v2y with integral coefficients, there exist arbitrarily large finite sets of integers A and B such that |f(A)| > |g(A)| and |f...
متن کاملInverse Problems for Linear Forms over Finite Sets of Integers
Let f(x1, x2, . . . , xm) = u1x1 + u2x2 + · · · + umxm be a linear form with positive integer coefficients, and let Nf (k) = min{|f(A)| : A ⊆ Z and |A| = k}. A minimizing k-set for f is a set A such that |A| = k and |f(A)| = Nf (k). A finite sequence (u1, u2, . . . , um) of positive integers is called complete if n
متن کاملLinear Forms and Complementing Sets of Integers
Let φ(x1, . . . , xh, y) = u1x1 + · · · + uhxh + vy be a linear form with nonzero integer coefficients u1, . . . , uh, v. Let A = (A1, . . . , Ah) be an h-tuple of finite sets of integers and let B be an infinite set of integers. Define the representation function associated to the form φ and the sets A and B as follows: R (φ) A,B (n) = card ({(a1, . . . , ah, b) ∈ A1 × · · · × Ah × B : φ(a1, ....
متن کاملOn iterating linear transformations over recognizable sets of integers
It has been known for a long time that the sets of integer vectors that are recognizable by finite-state automata are those that can be defined in an extension of Presburger arithmetic. In this paper, we address the problem of deciding whether the closure of a linear transformation preserves the recognizable nature of sets of integer vectors. We solve this problem by introducing an original ext...
متن کاملRepresentations of Integers by Linear Forms in Nonnegative Integers*
Let Sz be the set of positive integers that are omitted values of the form f = z”= a.x. where the a, are fixed and relatively prime natural numbers *1 $1) and the xi are variable nonnegative integers. Set w = #Q and K = max 0 + 1 (the conductor). Properties of w and K are studied, such as an estimate for w (similar to one found by Brauer) and the inequality 2w > K. The so-called Gorenstein cond...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2007
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa129-4-5