A Detailed Approach to Calculating Lattice Points in Triangles

The problem of calculating the total number of lattice points in a triangle with vertices at (0,0), (N,0), and (N,Ne), where e is an irrational number, is a beautiful and fascinating problem in combinatorial geometry. Despite the challenge posed by the non-rational nature of e, we can leverage powerful mathematical theorems and concepts to find a solution.

Introduction to Lattice Points and Triangles

A lattice point in the context of this problem refers to points with integer coordinates. Given the triangle with vertices at (0,0), (N,0), and (N,Ne), the question is how to count the lattice points more efficiently than simply evaluating the summation directly.

Pick's Theorem and Its Limitations

One of the most valuable tools for this purpose is Pick's Theorem, which relates the number of lattice points inside or on a lattice polygon to its area. However, the triangle we are dealing with is not a lattice polygon, as the point (N,Ne) does not have rational coordinates due to e being irrational. This impasse can be overcome by a method called the Shoelace method with Pick's Theorem, which we will explore in detail below.

Shoehorning Pick's Theorem: A Modified Approach

To address the issue of Pick's Theorem not directly applying, we can create a series of lattice polygons that enclose all the required lattice points. Specifically, we will construct a polygon with vertices that include the point (N,Ne) as an interior point. Using the point (N,Ne) as reference, we can define a sequence of lattice points that approximate the behavior of (N,Ne).

For each integer ( n leq N ) such that ( frac{n}{m} ) approximates ( e ) tightly, we define a point ( P_i m_i n_i ) where ( n_i ) and ( m_i ) are chosen such that the difference ( frac{n_i}{m_i} - e ) is minimized. This process continues until we reach the point ( P_k N lceil eN rceil ).

The set of interior points of the polygon with vertices ((0,0), P_1, P_2, ldots, P_k, (N,0)) is exactly the set of desired points. This is because the polygon contains the triangle mentioned above and no other points within the polygon are above the line ( y ex ).

Constructing the Lattice Points: Continued Fraction Approximation

To construct these points, we leverage continued fraction approximation, a powerful tool in number theory. The continued fraction expansion of ( e ) provides a sequence of rational approximations ( frac{p_i}{q_i} ) to ( e ) such that ( frac{p_i}{q_i} ) is the closest rational number to ( e ) with denominator ( q_i ).

Starting with ( P_1 ), we find the best approximation to ( e ) with a denominator less than a certain threshold. Then, for each subsequent ( P_i ), we choose the largest ( j ) such that ( n_{i-1} leq q_j ) and set ( P_i m_{i-1} p_j n_{i-1} q_j ). This iterative process ensures that we get at most ( O(log N) ) points.

This method guarantees that we can find a sequence of points that closely approximates the point (N, Ne), allowing us to apply Pick's Theorem to the enlarged polygon.

Conclusion and Further Reading

The problem of lattice point enumeration in triangles, while challenging, is rich in mathematical beauty. By using concepts such as continued fractions and Pick's Theorem, we can effectively solve this problem. The Ehrhart polynomial is another fascinating topic in this field, offering a deeper understanding of lattice point enumeration in more complex geometric shapes.

For further exploration, I highly recommend delving into the combinatorial properties of lattice points and their applications in various geometric and number-theoretic problems.